Surface acoustic wave resonator, surface acoustic wave oscillator, and electronic apparatus

ABSTRACT

A surface acoustic wave resonator includes a quartz substrate with preselected Euler angles and an IDT on the quartz substrate. The IDT includes electrode fingers and excites a stop band upper end mode surface acoustic wave. Inter-electrode finger grooves are provided between the electrode fingers. Assuming a surface acoustic wave wavelength is λ, an electrode finger film thickness is H, an inter-electrode finger groove depth is G, a line occupation rate of convex portions of the substrate between the inter-electrode finger grooves is ηg, and a line occupation rate of the electrode fingers on the convex portions is ηe, 0.0407λ≦G+H; and ηg&gt;ηe.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a continuation application of U.S. application Ser. No.13/310,123 filed Dec. 2, 2011 which claims priority to Japanese PatentApplication No. 2010-270953 filed Dec. 3, 2010 all of which areincorporated by reference herein in their entireties.

BACKGROUND

1. Technical Field

The present invention relates to a surface acoustic wave resonator, asurface acoustic wave oscillator in which the resonator is mounted, andan electronic apparatus, and more particularly to a type of surfaceacoustic wave resonator where grooves are provided in a substratesurface, and a surface acoustic wave oscillator in which the resonatoris mounted.

2. Related Art

In a surface acoustic wave (SAW) device (for example, a SAW resonator),the effect of a SAW stop band, piezoelectric substrate (for example,quartz crystal substrate) cut angle, IDT (interdigital transducer)formation shape, and the like, on changes in frequency-temperaturecharacteristics is considerable.

For example, a configuration exciting each of a SAW stop band upper endmode and lower end mode, the distribution of standing waves in each ofthe stop band upper end mode and lower end mode, and the like, aredisclosed in JP-A-11-214958.

In addition, points for which the SAW stop band upper end mode hasbetter frequency-temperature characteristics than the stop band lowerend mode are described in JP-A-2006-148622, JP-A-2007-208871,JP-A-2007-267033 and JP-A-2002-100959. Then, it is described inJP-A-2006-148622 and JP-A-2007-208871 that, in order to obtain favorablefrequency-temperature characteristics in a SAW device utilizing aRayleigh wave, as well as adjusting the cut angle of the quartz crystalsubstrate, the electrode standardizing film thickness (H/λ) is increasedto around 0.1.

Further, it is described in JP-A-2007-267033 that, as well as adjustingthe cut angle of the quartz crystal substrate in a SAW device utilizinga Rayleigh wave, the electrode standardizing film thickness (H/λ) isincreased by around 0.045 or more.

In addition, it is described in JP-A-2002-100959 that, by using arotated Y-cut, X-propagating quartz crystal substrate, and utilizing thestop band upper end resonance, the frequency-temperature characteristicsimprove more than in the case of using the stop band lower endresonance.

It is described in JP-A-57-5418 and “Manufacturing Conditions andCharacteristics of Grooved SAW Resonators” (Institute of Electronics andCommunication Engineers of Japan Technical Research Report MW82-59(1982)), that grooves are provided between the electrode fingersconfiguring the IDT, and between the conductor strips configuring thereflectors, in a SAW device using an ST cut quartz crystal substrate.Also, it is described in “Manufacturing Conditions and Characteristicsof Grooved SAW Resonators” (Institute of Electronics and CommunicationEngineers of Japan Technical Research Report MW82-59 (1982)), that apeak temperature in the frequency-temperature characteristics having aquadratic curve shape changes depending on the depth of the grooves, anda second-order temperature coefficient is approximately −3.4×10⁻⁸/° C.².

In Japanese Patent No. 3,851,336, as well as describing a configurationfor making a curve indicating the frequency-temperature characteristicsa cubic curve in a SAW device using an LST cut quartz crystal substrate,it is described that, in a SAW device using a Rayleigh wave, it has notbeen possible to find a cut angle substrate having the kind oftemperature characteristics indicated by a cubic curve.

As described above, there is a wide range of elements for improving thefrequency-temperature characteristics, and it is thought that,particularly with a SAW device using a Rayleigh wave, increasing thefilm thickness of the electrodes configuring the IDT is one factorcontributing to the frequency-temperature characteristics. However, theapplicant has found experimentally that on increasing the film thicknessof the electrodes, environmental resistance characteristics, such astemporal change characteristics and temperature and shock resistancecharacteristics, are deteriorated. Further, when having the improvementof frequency-temperature characteristics as a principal object, it isnecessary to increase the electrode film thickness, as described above,and an accompanying deterioration of temporal change characteristics,temperature and shock resistance characteristics, and the like, isunavoidable. Since this also applies to the Q value, it is difficult torealize a higher Q without increasing the electrode film thickness.

In order to solve the problem, Pamphlet of International PublicationWO2010/098139 discloses a configuration in which grooves are formed on aquartz crystal substrate in a direction perpendicular to the propagationdirection of the surface acoustic wave, and electrodes are formed onconvex portions formed by the grooves. With this, environmentalresistance characteristics such as temporal change characteristics andtemperature and shock resistance characteristics are improved, therebyrealizing a high Q value. In addition, JP-A-61-220513 or JP-A-61-220514discloses a configuration in which, in order to realize a high Q value,grooves are formed between stripe-shaped metal films constitutingreflectors which are disposed between IDT electrodes or at both sides ofthe IDT electrode.

Further, in Pamphlet of International Publication WO2010/098139, agroove depth, a film thickness of an electrode formed on the groove, anda line occupation rate of the groove are systematically investigated. Inaddition, in a case where the surface acoustic wave resonator is excitedin a stop band upper end mode, the condition is ascertained that anabsolute value of the second-order temperature coefficient of thesurface acoustic wave is 0.01 ppm/° C.² or less by adjusting the lineoccupation rate with respect to a given groove depth and electrode filmthickness. Thereby, since the frequency-temperature characteristics ofthe surface acoustic wave show a cubic curve, it is expected that afrequency deviation can be suppressed in a temperature range around theinflection point.

In addition, JP-A-2009-225420 discloses a configuration in which afrequency deviation in an operating temperature range of the surfaceacoustic wave resonator is reduced when the line width of the electrodefinger constituting the IDT electrode, that is, a line occupation ratefluctuates.

However, there is a strong demand for reduction in loss in the surfaceacoustic wave resonator in the surface acoustic wave resonator disclosedin Pamphlet of International Publication WO2010/098139, JP-A-61-220513,JP-A-61-220514, and JP-A-2009-225420 as well, but there is no detaileddisclosure thereof at present.

SUMMARY

An advantage of some aspects of the invention is to provide a surfaceacoustic wave resonator which reduces a frequency deviation of a surfaceacoustic wave and decreases loss in the surface acoustic wave resonator,and a surface acoustic wave oscillator and an electronic apparatus usingthe surface acoustic wave resonator.

Application Example 1

This application example of the invention is directed to a surfaceacoustic wave resonator including a quartz crystal substrate with Eulerangles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and 42.79°≦|Ψ|≦49.57°); and an IDTthat is provided on the quartz crystal substrate, includes a pluralityof electrode fingers, and excites a stop band upper end mode surfaceacoustic wave, wherein inter-electrode finger grooves which aredepressions of the quartz crystal substrate are provided between theelectrode fingers in a plan view, and wherein if a line occupation rateof convex portions of the quartz crystal substrate disposed between theinter-electrode finger grooves is ηg, and a line occupation rate of theelectrode fingers disposed on the convex portions is ηe, the followingrelationships are satisfied in a case where an effective line occupationrate ηeff of the IDT is an arithmetic mean of the line occupation rateηg and the line occupation rate ηe.η_(g)>η_(e),0.59<η_(eff)<0.73

With the configuration, it is possible to reduce loss in a surfaceacoustic wave resonator by increasing excitation efficiency, and tosuppress a fluctuation amount of the first-order temperature coefficientof the surface acoustic wave resonator so as to suppress variations inthe resonance frequency.

Application Example 2

This application example of the invention is directed to the surfaceacoustic wave resonator according to the above application example ofthe invention, wherein, if a wavelength of the surface acoustic wave isλ, a depth of the inter-electrode finger groove is G, an electrode filmthickness of the IDT is H, and plane coordinates (G/λ, ηeff) areindicated using the value G/λ obtained by dividing a depth G of theinter-electrode finger groove by the wavelength λ of the surfaceacoustic wave, and the effective line occupation rate ηeff, the planecoordinates (G/λ, ηeff) are included in one of

(1) at 0.000λ<H≦0.005λ

a range surrounded by a line connecting (0.010, 0.710), (0.020, 0.710),(0.030, 0.710), (0.040, 0.710), (0.050, 0.710), (0.060, 0.710), (0.070,0.710), (0.080, 0.710), (0.090, 0.710), (0.090, 0.420), (0.080, 0.570),(0.070, 0.590), (0.060, 0.615), (0.050, 0.630), (0.040, 0.635), (0.030,0.650), (0.020, 0.670), and (0.010, 0.710) in this order, anda range surrounded by a line connecting (0.030, 0.590), (0.040, 0.580),(0.050, 0.550), (0.060, 0.520), (0.070, 0.480), (0.080, 0.450), (0.090,0.400), (0.090, 0.180), (0.080, 0.340), (0.070, 0.410), (0.060, 0.460),(0.050, 0.490), (0.040, 0.520), (0.030, 0.550), and (0.030, 0.590) inthis order,(2) at 0.005λ<H≦0.010λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.740),(0.030, 0.715), (0.040, 0.730), (0.050, 0.740), (0.060, 0.730), (0.070,0.730), (0.080, 0.730), (0.080, 0.500), (0.070, 0.570), (0.060, 0.610),(0.050, 0.630), (0.040, 0.635), (0.030, 0.655), (0.020, 0.680), (0.010,0.760), and (0.010, 0.770) in this order, anda range surrounded by a line connecting (0.020, 0.650), (0.030, 0.610),(0.040, 0.570), (0.050, 0.550), (0.060, 0.520), (0.070, 0.470), (0.070,0.370), (0.060, 0.440), (0.050, 0.480), (0.040, 0.520), (0.030, 0.550),(0.020, 0.590), and (0.020, 0.650) in this order,(3) at 0.010λ<H≦0.015λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.760),(0.030, 0.760), (0.040, 0.750), (0.050, 0.750), (0.060, 0.750), (0.070,0.740), (0.080, 0.740), (0.080, 0.340), (0.070, 0.545), (0.060, 0.590),(0.050, 0.620), (0.040, 0.645), (0.030, 0.670), (0.020, 0.705), (0.010,0.760), and (0.010, 0.770) in this order, anda range surrounded by a line connecting (0.010, 0.740), (0.020, 0.650),(0.030, 0.610), (0.040, 0.570), (0.050, 0.540), (0.060, 0.480), (0.070,0.430), (0.070, 0.350), (0.060, 0.420), (0.050, 0.470), (0.040, 0.510),(0.030, 0.550), (0.020, 0.610), (0.010, 0.700), and (0.010, 0.740) inthis order,(4) at 0.015λ<H≦0.020λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.770),(0.030, 0.760), (0.040, 0.760), (0.050, 0.760), (0.060, 0.750), (0.070,0.750), (0.070, 0.510), (0.060, 0.570), (0.050, 0.620), (0.040, 0.640),(0.030, 0.660), (0.020, 0.675), (0.010, 0.700), and (0.010, 0.770) inthis order, anda range surrounded by a line connecting (0.010, 0.690), (0.020, 0.640),(0.030, 0.590), (0.040, 0.550), (0.050, 0.510), (0.060, 0.470), (0.070,0.415), (0.070, 0.280), (0.060, 0.380), (0.050, 0.470), (0.040, 0.510),(0.030, 0.550), (0.020, 0.610), (0.010, 0.680), and (0.010, 0.690) inthis order,(5) at 0.020λ<H≦0.025λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.770),(0.030, 0.760), (0.040, 0.760), (0.050, 0.760), (0.060, 0.760), (0.070,0.760), (0.070, 0.550), (0.060, 0.545), (0.050, 0.590), (0.040, 0.620),(0.030, 0.645), (0.020, 0.680), (0.010, 0.700), and (0.010, 0.770) inthis order, anda range surrounded by a line connecting (0.010, 0.690), (0.020, 0.640),(0.030, 0.590), (0.040, 0.550), (0.050, 0.510), (0.060, 0.420), (0.070,0.415), (0.070, 0.340), (0.060, 0.340), (0.050, 0.420), (0.040, 0.470),(0.030, 0.520), (0.020, 0.580), (0.010, 0.650), and (0.010, 0.690) inthis order,(6) at 0.025λ<H≦0.030λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.770),(0.030, 0.770), (0.040, 0.760), (0.050, 0.760), (0.060, 0.760), (0.070,0.760), (0.070, 0.550), (0.060, 0.505), (0.050, 0.590), (0.040, 0.620),(0.030, 0.645), (0.020, 0.680), (0.010, 0.700), and (0.010, 0.770) inthis order, anda range surrounded by a line connecting (0.010, 0.670), (0.020, 0.605),(0.030, 0.560), (0.040, 0.520), (0.050, 0.470), (0.060, 0.395), (0.070,0.500), (0.070, 0.490), (0.060, 0.270), (0.050, 0.410), (0.040, 0.470),(0.030, 0.520), (0.020, 0.580), (0.010, 0.620), and (0.010, 0.670) inthis order(7) at 0.030λ<H≦0.035λa range surrounded by a line connecting (0.010, 0.770), (0.020, 0.770),(0.030, 0.770), (0.040, 0.760), (0.050, 0.760), (0.060, 0.760), (0.070,0.760), (0.070, 0.550), (0.060, 0.500), (0.050, 0.545), (0.040, 0.590),(0.030, 0.625), (0.020, 0.650), (0.010, 0.680), and (0.010, 0.770) inthis ordera range surrounded by a line connecting (0.010, 0.655), (0.020, 0.590),(0.030, 0.540), (0.040, 0.495), (0.050, 0.435), (0.060, 0.395), (0.070,0.500), (0.070, 0.550), (0.060, 0.380), (0.050, 0.330), (0.040, 0.410),(0.030, 0.470), (0.020, 0.520), (0.010, 0.590), and (0.010, 0.655) inthis order.

With the configuration, it is possible to suppress an absolute value ofthe second-order temperature coefficient of the surface acoustic waveresonator to 0.01 ppm/° C.² or less so as to correspond to the thicknessof H.

Application Example 3

This application example of the invention is directed to the surfaceacoustic wave resonator according to the above application example ofthe invention, wherein the depth G of the inter-electrode finger grooveand the effective line occupation rate ηeff satisfy the followingrelationships.−2.0000×G/λ+0.7200≦η_(eff)−2.5000×G/λ+0.7775 provided that0.0100λ≦G≦0.0500%−3.5898×G/λ+0.7995≦η_(eff)≦−2.5000×G/λ+0.7775 provided that0.0500λ<G≦0.0695λ

With the configuration, it is possible to suppress an absolute value ofthe second-order temperature coefficient of the surface acoustic waveresonator to 0.01 ppm/° C.² or less.

Application Example 4

This application example of the invention is directed to the surfaceacoustic wave resonator according to Application Example 3, wherein theelectrode film thickness H of the IDT satisfies the followingrelationship.0<H≦0.035λ

With the surface acoustic wave resonator having such a characteristic,it is possible to realize favorable frequency-temperaturecharacteristics in the operating temperature range. In addition, withthe characteristics, it is possible to suppress deterioration inenvironmental resistance characteristics due to the increase in theelectrode film thickness.

Application Example 5

This application example of the invention is directed to the surfaceacoustic wave resonator according to Application Example 4, wherein theeffective line occupation rate ηeff satisfies the followingrelationship.

η_(eff) = −1963.05 × (G/λ)³ + 196.28 × (G/λ)² − 6.53 × (G/λ) − 135.99 × (H/λ)² + 5.817 × (H/λ) + 0.732 − 99.99 × (G/λ) × (H/λ) ± 0.04

By fixing ηeff to satisfy the above Equation in the electrode filmthickness range in Application Example 4, it is possible to suppress anabsolute value of the second-order temperature coefficient to 0.01 ppm/°C.² or less.

Application Example 6

This application example of the invention is directed to the surfaceacoustic wave resonator according to any one of Application Examples 2,4 and 5, wherein the sum of the depth G of the inter-electrode fingergroove and the electrode film thickness H satisfies the followingrelationship.0.0407λ≦G+H

By setting the sum of the depth G of the inter-electrode finger grooveand the electrode film thickness H as in the above Expression, it ispossible to obtain a higher Q value than that of the surface acousticwave resonator in the related art.

Application Example 7

This application example of the invention is directed to the surfaceacoustic wave resonator according to any one of Application Examples 1to 6, wherein Ψ and θ satisfy the following relationship.ψ=1.191×10⁻³×θ³−4.490×10⁻¹×θ²+5.646×10¹×θ−2.324×10³±1.0

By manufacturing a surface acoustic wave resonator using a quartzcrystal substrate which is cut with the cut angle having thecharacteristics, it is possible to realize a surface acoustic waveresonator showing favorable frequency-temperature characteristics in awide range.

Application Example 8

This application example of the invention is directed to the surfaceacoustic wave resonator according to any one of Application Example 1 to7, wherein, if a stop band upper end frequency of the IDT is ft2, a stopband lower end mode frequency of reflectors provided so as to sandwichthe IDT in the propagation direction of the surface acoustic wave isfr1, and a stop band upper end mode frequency of the reflectors is fr2,the following relationship is satisfied.fr1<ft2<fr2

With the characteristics, the reflection coefficient |Γ| of thereflectors increases in the stop band upper end mode frequency ft2 ofthe IDT, and thus the stop band upper end mode surface acoustic waveexcited from the IDT is reflected to the IDT side by the reflectors at ahigh reflection coefficient. Then, energy confinement of the stop bandupper end mode surface acoustic wave becomes strong, and thereby it ispossible to realize a low-loss surface acoustic wave resonator.

Application Example 9

This application example of the invention is directed to the surfaceacoustic wave resonator according to any one of Application Examples 1to 8, an inter-conductor strip groove is provided between conductorstrips forming the reflectors, wherein a depth of the inter-conductorstrip groove is smaller than that of the inter-electrode finger groove.

With the characteristics, it is possible to shift the reflector stopband to a higher frequency than the IDT stop band. Therefore, it ispossible to realize the relationship of the equation described inApplication Example 8.

Application Example 10

This application example of the invention is directed to a surfaceacoustic wave oscillator including the surface acoustic wave resonatoraccording to any one of Application Examples 1 to 9 and a circuit thatdrives the IDT.

Application Example 11

This application example of the invention is directed to an electronicapparatus including the surface acoustic wave resonator according to anyone of Application Examples 1 to 9.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the accompanyingdrawings, wherein like numbers reference like elements.

FIGS. 1A to 1D are diagrams illustrating a configuration of a SAW deviceaccording to a first embodiment, where FIG. 1A is a plan view of theconfiguration, FIG. 1B is a partial enlarged sectional view of a sidesurface, FIG. 1C is a partial enlarged view for describing details ofFIG. 1B, and FIG. 1D, being the partial enlarged view of FIG. 1C, showsa cross-sectional shape of a groove portion conceivable whenmanufacturing a SAW resonator using a photolithography technique and anetching technique.

FIG. 2 is a diagram illustrating an example of the orientation of awafer which is a base material of a quartz crystal substrate used in theinvention.

FIGS. 3A and 3B are diagrams illustrating configuration examples of theSAW device when employing a tilted type IDT as a modified example of thefirst embodiment, where FIG. 3A is an example of an appearance oftilting electrode fingers, making them perpendicular to an X″ axis, andFIG. 3B is an example of the SAW device having an IDT in which bus barsconnecting the electrode fingers are tilted.

FIG. 4 is a diagram illustrating a relationship between a stop bandupper end mode and lower end mode.

FIG. 5 is a graph illustrating a relationship between the depth of aninter-electrode finger groove and a frequency fluctuation amount in anoperating temperature range.

FIG. 6 is a diagram illustrating temperature characteristics in an STcut quartz crystal substrate.

FIGS. 7A to 7D are graphs illustrating differences in a change in asecond-order temperature coefficient accompanying a change in a lineoccupation rate η at a stop band upper end mode resonance point and astop band lower end mode resonance point, where FIG. 7A is a graphillustrating a displacement of a stop band upper end mode second-ordertemperature coefficient β when a groove depth G is 2% λ, FIG. 7B is agraph illustrating a displacement of a stop band lower end modesecond-order temperature coefficient β when the groove depth G is 2% λ,FIG. 7C is a graph illustrating a displacement of the stop band upperend mode second-order temperature coefficient β when the groove depth Gis 4% λ, and FIG. 7D is a graph illustrating a displacement of the stopband lower end mode second-order temperature coefficient β when thegroove depth G is 4% λ.

FIGS. 8A to 8I are graphs illustrating a relationship between the lineoccupation rate η and a second-order temperature coefficient β whenchanging the inter-electrode finger groove depth, with an electrode filmthickness as 0, where FIG. 8A is a graph when the groove depth G is 1%λ, FIG. 8B is a graph when the groove depth G is 1.25% λ, FIG. 8C is agraph when the groove depth G is 1.5% λ, FIG. 8D is a graph when thegroove depth G is 2% λ, FIG. 8E is a graph when the groove depth G is 3%λ, FIG. 8F is a graph when the groove depth G is 4% λ, FIG. 8G is agraph when the groove depth G is 5% λ, FIG. 8H is a graph when thegroove depth G is 6% λ, and FIG. 8I is a graph when the groove depth Gis 8% λ.

FIG. 9 is a graph illustrating a relationship between an inter-electrodefinger groove depth at which the second-order temperature coefficientbecomes 0 and the line occupation rate η, when the electrode filmthickness is 0.

FIGS. 10A to 10I are graphs illustrating a relationship between the lineoccupation rate η and a frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0, where FIG. 10A is a graph when the groove depth G is 1%λ, FIG. 10B is a graph when the groove depth G is 1.25% λ, FIG. 10C is agraph when the groove depth G is 1.5% λ, FIG. 10D is a graph when thegroove depth G is 2% λ, FIG. 10E is a graph when the groove depth G is3% λ, FIG. 10F is a graph when the groove depth G is 4% λ, FIG. 10G is agraph when the groove depth G is 5% λ, FIG. 10H is a graph when thegroove depth G is 6% λ, and FIG. 10I is a graph when the groove depth Gis 8% λ.

FIG. 11 is a graph illustrating a relationship between theinter-electrode finger groove depth and the frequency fluctuation amountwhen the inter-electrode finger groove depth deviates by ±0.001λ.

FIGS. 12A to 12F are graphs illustrating a relationship between theinter-electrode finger groove depth at which the second-ordertemperature coefficient becomes 0 and the line occupation rate η, whenthe electrode film thickness is changed, where FIG. 12A is a graph whenthe electrode film thickness is 1% λ, FIG. 12B is a graph when theelectrode film thickness is 1.5% λ, FIG. 12C is a graph when theelectrode film thickness is 2% λ, FIG. 12D is a graph when the electrodefilm thickness is 2.5% λ, FIG. 12E is a graph when the electrode filmthickness is 3% λ, and FIG. 12F is a graph when the electrode filmthickness is 3.5% λ.

FIGS. 13A to 13B are diagrams in which relationships between η1 at whichthe second-order temperature coefficient β≅0 (ppm/° C.²) and theinter-electrode finger groove depth for each electrode film thicknessare summarized in graphs, where FIG. 13A shows a relationship betweenthe groove depth G and η1 when changing the electrode film thicknessfrom 1% λ, to 3.5% λ, and FIG. 13B is a diagram proving that an area inwhich |β|≦0.01 (ppm/° C.²) is inside a polygon formed by linking pointsA to H.

FIG. 14 is a diagram illustrating in an approximate curve a relationshipbetween the inter-electrode finger groove depth and line occupation rateη for electrode film thicknesses from H≅0 to H=0.035λ.

FIGS. 15A to 15F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.01λ, where FIG. 15A is a graph when thegroove depth G is 0, FIG. 15B is a graph when the groove depth G is 1%λ, FIG. 15C is a graph when the groove depth G is 2% λ, FIG. 15D is agraph when the groove depth G is 3% λ, FIG. 15E is a graph when thegroove depth G is 4% λ, and FIG. 15F is a graph when the groove depth Gis 5% λ.

FIGS. 16A to 16F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.015λ, where FIG. 16A is a graph when thegroove depth G is 0, FIG. 16B is a graph when the groove depth G is 1%λ, FIG. 16C is a graph when the groove depth G is 1.5% λ, FIG. 16D is agraph when the groove depth G is 2.5% λ, FIG. 16E is a graph when thegroove depth G is 3.5% λ, and FIG. 16F is a graph when the groove depthG is 4.5% λ.

FIGS. 17A to 17F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.02λ, where FIG. 17A is a graph when thegroove depth G is 0, FIG. 17B is a graph when the groove depth G is 1%λ, FIG. 17C is a graph when the groove depth G is 2% λ, FIG. 17D is agraph when the groove depth G is 3% λ, FIG. 17E is a graph when thegroove depth G is 4% λ, and FIG. 17F is a graph when the groove depth Gis 5% λ.

FIGS. 18A to 18F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.025λ, where FIG. 18A is a graph when thegroove depth G is 0, FIG. 18B is a graph when the groove depth G is 1%λ, FIG. 18C is a graph when the groove depth G is 1.5% λ, FIG. 18D is agraph when the groove depth G is 2.5% λ, FIG. 18E is a graph when thegroove depth G is 3.5% λ, and FIG. 18F is a graph when the groove depthG is 4.5% λ.

FIGS. 19A to 19F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.03λ, where FIG. 19A is a graph when thegroove depth G is 0, FIG. 19B is a graph when the groove depth G is 1%λ, FIG. 19C is a graph when the groove depth G is 2% λ, FIG. 19D is agraph when the groove depth G is 3% λ, FIG. 19E is a graph when thegroove depth G is 4% λ, and FIG. 19F is a graph when the groove depth Gis 5% λ.

FIGS. 20A to 20F are graphs illustrating the relationship between theline occupation rate η and the second-order temperature coefficient βwhen changing the inter-electrode finger groove depth, with theelectrode film thickness as 0.035λ, where FIG. 20A is a graph when thegroove depth G is 0, FIG. 20B is a graph when the groove depth G is 1%λ, FIG. 20C is a graph when the groove depth G is 2% λ, FIG. 20D is agraph when the groove depth G is 3% λ, FIG. 20E is a graph when thegroove depth G is 4% λ, and FIG. 20F is a graph when the groove depth Gis 5% λ.

FIGS. 21A to 21F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.01λ, where FIG. 21A is a graph when the groove depth G is0, FIG. 21B is a graph when the groove depth G is 1% λ, FIG. 21C is agraph when the groove depth G is 2% λ, FIG. 21D is a graph when thegroove depth G is 3% λ, FIG. 21E is a graph when the groove depth G is4% λ, and FIG. 21F is a graph when the groove depth G is 5% λ.

FIGS. 22A to 22F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.015λ, where FIG. 22A is a graph when the groove depth Gis 0, FIG. 22B is a graph when the groove depth G is 1% λ, FIG. 22C is agraph when the groove depth G is 1.5% λ, FIG. 22D is a graph when thegroove depth G is 2.5% λ, FIG. 22E is a graph when the groove depth G is3.5% λ, and FIG. 22F is a graph when the groove depth G is 4.5% λ.

FIGS. 23A to 23F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.02λ, where FIG. 23A is a graph when the groove depth G is0, FIG. 23B is a graph when the groove depth G is 1% λ, FIG. 23C is agraph when the groove depth G is 2% λ, FIG. 23D is a graph when thegroove depth G is 3% λ, FIG. 23E is a graph when the groove depth G is4% λ, and FIG. 23F is a graph when the groove depth G is 5% λ.

FIGS. 24A to 24F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.025λ, where FIG. 24A is a graph when the groove depth Gis 0, FIG. 24B is a graph when the groove depth G is 1% λ, FIG. 24C is agraph when the groove depth G is 1.5% λ, FIG. 24D is a graph when thegroove depth G is 2.5% λ, FIG. 24E is a graph when the groove depth G is3.5% λ, and FIG. 24F is a graph when the groove depth G is 4.5% λ.

FIGS. 25A to 25F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.03λ, where FIG. 25A is a graph when the groove depth G is0, FIG. 25B is a graph when the groove depth G is 1% λ, FIG. 25C is agraph when the groove depth G is 2% λ, FIG. 25D is a graph when thegroove depth G is 3% λ, FIG. 25E is a graph when the groove depth G is4% λ, and FIG. 25F is a graph when the groove depth G is 5% λ.

FIGS. 26A to 26F are graphs illustrating the relationship between theline occupation rate η and frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.035λ, where FIG. 26A is a graph when the groove depth Gis 0, FIG. 26B is a graph when the groove depth G is 1% λ, FIG. 26C is agraph when the groove depth G is 2% λ, FIG. 26D is a graph when thegroove depth G is 3% λ, FIG. 26E is a graph when the groove depth G is4% λ, and FIG. 26F is a graph when the groove depth G is 5% λ.

FIGS. 27A and 27B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0≦H<0.005λ, where FIG. 27A shows a case of η1, and FIG.27B shows a case of η2.

FIGS. 28A and 28B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.05λ≦H<0.010λ, where FIG. 28A shows a case of η1, andFIG. 28B shows a case of η2.

FIGS. 29A and 29B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.010λ≦H<0.015λ, where FIG. 29A shows a case of η1, andFIG. 29B shows a case of η2.

FIGS. 30A and 30B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.015λ≦H<0.020λ, where FIG. 30A shows a case of η1, andFIG. 30B shows a case of η2.

FIGS. 31A and 31B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.020λ≦H<0.025λ, where FIG. 31A shows a case of η1, andFIG. 31B shows a case of η2.

FIGS. 32A and 32B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.025λ≦H<0.030λ, where FIG. 32A shows a case of η1, andFIG. 32B shows a case of η2.

FIGS. 33A and 33B are diagrams illustrating a range in which |β|≦0.01(ppm/° C.²) by means of graphs illustrating a relationship between theline occupation rate η and groove depth G when the electrode filmthickness H is 0.030λ≦H<0.035λ, where FIG. 33A shows a case of η1, andFIG. 33B shows a case of η2.

FIGS. 34A to 34F are graphs illustrating relationships between theinter-electrode finger groove depth and an Euler angle Ψ when theelectrode film thickness and line occupation rate η (η1 solid line, η2:broken line) are fixed, where FIG. 34A is a graph when the electrodefilm thickness is 1% λ, FIG. 34B is a graph when the electrode filmthickness is 1.5% λ, FIG. 34C is a graph when the electrode filmthickness is 2% λ, FIG. 34D is a graph when the electrode film thicknessis 2.5% λ, FIG. 34E is a graph when the electrode film thickness is 3%λ, and FIG. 34F is a graph when the electrode film thickness is 3.5% λ.

FIG. 35 is a diagram in which the relationships between theinter-electrode finger groove depth G and Euler angle Ψ for eachelectrode film thickness H are summarized in a graph.

FIG. 36 is a graph illustrating a relationship between aninter-electrode finger groove depth at which the second-ordertemperature coefficient β is −0.01 (ppm/° C.²) and the Euler angle Ψ.

FIG. 37 is a graph illustrating a relationship between aninter-electrode finger groove depth at which the second-ordertemperature coefficient β is +0.01 (ppm/° C.²) and the Euler angle Ψ.

FIGS. 38A and 38B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0<H≦0.005λ, where FIG. 38A is a graph illustrating amaximum value and minimum value of Ψ, and FIG. 38B is a graph of an areaof Ψ which satisfies the requirement of β.

FIGS. 39A and 39B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.005λ<H≦0.010μ, where FIG. 39A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 39B is agraph of an area of Ψ which satisfies the requirement of β.

FIGS. 40A and 40B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.010λ<H≦0.015λ, where FIG. 40A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 40B is agraph of an area of Ψ which satisfies the requirement of β.

FIGS. 41A and 41B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.015λ<H≦0.020λ, where FIG. 41A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 41B is agraph of an area of Ψ which satisfies the requirement of β.

FIGS. 42A and 42B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.020λ<H≦0.025λ, where FIG. 42A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 42B is agraph of an area of Ψ which satisfies the requirement of β.

FIGS. 43A and 43B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.025λ<H≦0.030λ, where FIG. 43A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 43B is agraph of an area of Ψ which satisfies the requirement of β.

FIGS. 44A and 44B are graphs illustrating a range of Ψ which satisfiesthe requirement of |β|≦0.01 (ppm/° C.²) when the range of the electrodefilm thickness H is 0.030λ<H≦0.035λ, where FIG. 44A is a graphillustrating a maximum value and minimum value of Ψ, and FIG. 44B is agraph of an area of Ψwhich satisfies the requirement of β.

FIG. 45 is a graph illustrating a relationship between an Euler angle θand the second-order temperature coefficient β when the electrode filmthickness is 0.02λ and the inter-electrode finger groove depth is 0.04λ.

FIG. 46 is a graph illustrating a relationship between an Euler angle φand the second-order temperature coefficient β.

FIG. 47 is a graph illustrating a relationship between the Euler angle θand Euler angle Ψ when the frequency-temperature characteristics aregood.

FIG. 48 is a diagram illustrating an example of frequency-temperaturecharacteristics data from four test specimens under conditions at whichthe frequency-temperature characteristics are best.

FIG. 49 is a graph illustrating a relationship between a leveldifference, which is the sum of the inter-electrode finger groove depthand electrode film thickness, and a CI value.

FIG. 50 is a table illustrating an example of an equivalent circuitconstant and static characteristic of the SAW resonator according to theembodiment.

FIG. 51 is impedance curve data of the SAW resonator according to theembodiment.

FIG. 52 is a graph for comparing a relationship between the leveldifference and a Q value for the SAW resonator in the related art and arelationship between the level difference and the Q value for the SAWresonator according to the embodiment.

FIG. 53 is a diagram illustrating SAW reflection characteristics of theIDT and reflectors.

FIG. 54 is a graph illustrating a relationship between the electrodefilm thickness H and frequency fluctuation in a heat cycle test.

FIGS. 55A and 55B are diagrams illustrating a configuration of a SAWoscillator according to the embodiment.

FIGS. 56A and 56B are graphs illustrating frequency-temperaturecharacteristics of a SAW resonator, where FIG. 56A is a graphillustrating the frequency-temperature characteristics of the SAWresonator disclosed in JP-A-2006-203408, and FIG. 56B is a graphillustrating a range of the frequency-temperature characteristics in asubstantial operating temperature range.

FIG. 57 is a graph illustrating changes in the frequency fluctuationamount in the operating range for a SAW resonator of which the IDT andreflectors are covered with alumina as a protective film.

FIG. 58 is a graph illustrating changes in the frequency fluctuationamount in the operating range for a SAW resonator of which the IDT andreflectors are covered with SiO2 as a protective film.

FIGS. 59A to 59D are diagrams illustrating a configuration of a SAWresonator according to a second embodiment, where FIG. 59A is a planview of the configuration of the SAW resonator according to the secondembodiment, FIG. 59B is a partial enlarged sectional view of a sidesurface, FIG. 59C is an enlarged view for describing details of FIG.59B, and FIG. 59D, related to the partial enlarged view of FIG. 59C,shows a cross-sectional shape of a groove portion conceivable whenmanufacturing the SAW resonator according to the embodiment using aphotolithography technique and an etching technique, which is a diagramillustrating a method of specifying an effective line occupation rateηeff of an IDT electrode finger when the cross-sectional shape is notrectangular but trapezoidal.

FIG. 60 is a diagram illustrating frequency-temperature characteristicsof the type 1 and the type 2 in an example 1.

FIG. 61 is a diagram illustrating frequency-temperature characteristicsof the type 1 and the type 2 in an example 2.

FIG. 62 is a diagram illustrating changes in a fluctuation amount of thefirst-order temperature coefficient when the line occupation rate η ofthe type 1 is changed.

FIG. 63 is a diagram illustrating changes in a fluctuation amount of thefirst-order temperature coefficient when the effective line occupationrate ηeff of the type 2 is changed.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, a surface acoustic wave resonator and surface acoustic waveoscillator according to embodiments of the invention will be describedin detail with reference to the drawings. However, constituent elements,and kinds, combinations, shapes, relative dispositions thereof, and thelike are not intended to limit the scope of the invention unlessparticularly described but are only description examples.

First, a surface acoustic wave (SAW) resonator according to a firstembodiment of the invention will be described referring to FIGS. 1A to1D. In FIGS. 1A to 1D, FIG. 1A is a plan view of the SAW resonator, FIG.1B is a partial enlarged sectional view, FIG. 1C is an enlarged view fordescribing details of FIG. 1B, and FIG. 1D is a diagram which, relatedto the partial enlarged view of FIG. 1C, is for describing an IDTelectrode finger line occupation rate η identification method in a casewhere the cross-sectional shape is not rectangular but trapezoidal,which is a conceivable sectional shape when the SAW resonator accordingto the embodiment of the invention is manufactured using aphotolithography technique and an etching technique. It is appropriatethat the line occupation rate η is a proportion occupied by a width L ofa value (L+S), wherein a protrusion width L and a width S of a groove 32are added, at a height from the bottom of the groove 32 which is ½ of(G+H), which is a value where a depth (bump height) G of the groove 32and an electrode film thickness H are added.

The SAW resonator 10 according to the embodiment is basicallyconstituted by a quartz crystal substrate 30, an IDT 12, and reflectors20.

FIG. 2 is a diagram illustrating an orientation of a wafer 1, which is abase material of the quartz crystal substrate 30 used in the invention.In FIG. 2, an X axis is an electrical axis of the quartz crystal, a Yaxis is a mechanical axis of the quartz crystal, and a Z axis is anoptical axis of the quartz crystal. The wafer 1 has a cut surface with aZ′ axis as a normal, and has an X″ axis and a Y′″ axis perpendicular tothe X″ axis in the cut surface, as described later. Furthermore, the IDTand reflectors 20 configuring the SAW resonator 10 are disposed along anX″ axis in consideration of the propagation direction of the SAW, asdescribed later. The quartz crystal substrate 30 configuring the SAWresonator 10 is diced by being cut out of the wafer 1. Although theshape in a plan view of the quartz crystal substrate 30 is notparticularly limited, it may be a rectangle which has long sidesparallel to a X″ axis and has short sides parallel to the Y′″ axis.

In the embodiment, an in-plane rotation ST cut quartz crystal substrateexpressed by Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and42.79≦|Ψ|≦149.57°) is employed as the quartz crystal substrate 30. Here,a description will be made of the Euler angles with reference to FIG. 2.A substrate expressed by Euler angles (0°, 0°, and 0°) is a Z cutsubstrate 3 which has a main surface perpendicular to the Z axis.

Here, φ of Euler angles (φ, θ, and Ψ), relating to a first rotation ofthe Z cut substrate 3, is a first rotation angle, with the Z axis as arotation axis, and with a direction rotating from the +X axis to a +Yaxis side as a positive rotation angle. The X axis and Y axis after afirst rotation are respectively an X′ axis and a Y′ axis. In addition,in FIG. 2, as the description of the Euler angles, φ of the Euler anglesis indicated by 0°. Therefore, in FIG. 2, the X axis overlaps the X′axis, and the Y axis overlaps the Y′ axis.

The Euler angle θ, relating to a second rotation carried out after thefirst rotation of the Z cut substrate 3, is a second rotation angle,with the X axis (that is, the X′ axis) after the first rotation as arotation axis, and with a direction rotating from the +Y axis (that is,+Y′ axis) after the first rotation to the +Z axis as a positive rotationangle. The cut surface of a piezoelectric substrate, that is, the cutsurface of the above-described wafer 1 is determined by the firstrotation angle φ and the second rotation angle θ. That is to say, whenthe Y axis after the second rotation is a Y″ axis and the Z axis afterthe second rotation is a Z′ axis, a surface parallel to both the X′ axisand the Y″ axis is the cut surface of the piezoelectric substrate, andthe Z′ axis is a normal of the cut surface.

The Euler angle Ψ, relating to a third rotation carried out after thesecond rotation of the Z cut substrate 3, is a third rotation angle,with the Z′ axis which is the Z axis after the second rotation as arotation axis, and with a direction rotating, from the +X axis (that is,+X′ axis) after the second rotation to the +Y axis (that is, +Y″ axis)side after the second rotation as a positive rotation angle. Apropagation direction of the SAW is expressed by the third rotationangle Ψ with respect to the X axis (that is, the X′ axis) after thesecond rotation. That is to say, when the X axis after the thirdrotation is an X″ axis and the Y axis after the third rotation is a Y′″axis, a surface parallel to both the X″ axis and the Y′″ axis is the cutsurface of the piezoelectric substrate, and the Z′ axis is a normal ofthe cut surface. As such, since the normal does not vary even if thethird rotation is performed, the cut surface of the piezoelectricsubstrate is also the cut surface of the wafer 1. In addition, adirection parallel to the X″ axis is a propagation direction of the SAW.

Further, a SAW phase velocity direction is a direction parallel to theX″ axis direction. A phenomenon called a SAW power flow is a phenomenonin which a direction where a phase of the SAW progresses (phase velocitydirection) and a direction where energy of the SAW progresses (groupvelocity direction) are deviated from each other. An angle formed by thephase velocity direction and the group velocity direction is called apower flow angle (refer to FIGS. 3A and 3B).

The IDT 12 including a pair of pectinate electrodes 14 a and 14 b wherethe base end portions of a plurality of electrode fingers 18 a and 18 bare connected by bus bars 16 a and 16 b respectively, and the electrodefingers 18 a configuring one of the pectinate electrodes 14 a, and theelectrode fingers 18 b configuring the other pectinate electrode 14 b,are alternately disposed with a predetermined space between them.Furthermore, the electrode fingers 18 a and 18 b are disposed in such away that the extension direction of the electrode fingers 18 a and 18 bis perpendicular to the X″ axis, which is the propagation direction ofthe surface acoustic wave, as shown in FIG. 1A. A SAW excited by the SAWresonator 10 configured in this way, being a Rayleigh type SAW, hasoscillatory displacement elements on both the X″ axis and Z′ axis. Then,by displacing the SAW propagation direction from the X axis, which isthe crystal axis of the quartz crystal, in this way, it is possible toexcite a stop band upper end mode SAW.

In addition, furthermore, it is possible to give the SAW resonator 10according to a modified example of the first embodiment the kinds ofshape shown in FIGS. 3A and 3B. That is to say, even in a case ofapplying an IDT which is tilted by a power flow angle (hereinafter,referred to as a PFA) δ from the X″ axis, as shown in FIGS. 3A and 3B,it is possible to achieve a high Q by fulfilling the followingrequirements. FIG. 3A is a plan view illustrating an embodiment of atilted type IDT 12 a, the disposition conformation of the electrodefingers 18 a and 18 b in the tilted type IDT 12 a is tilted such thatthe X″ axis, which is the SAW propagation direction determined by theEuler angles, and the extension direction of the electrode fingers 18 aand 18 b of the tilted type IDT 12 a are in a perpendicularrelationship.

FIG. 3B is a plan view illustrating another embodiment of the tiltedtype IDT 12 a. In the example, the electrode finger array direction isdisposed tilted with respect to the X″ axis by tilting the bus bars 16 aand 16 b connecting the electrode fingers 18 a and 18 b respectively,but is configured such that the X″ axis and the extension direction ofthe electrode fingers 18 a and 18 b are in a perpendicular relationship,in the same way as in FIG. 3A.

Whichever kind of tilted type IDT is used, by disposing the electrodefingers in such a way that a direction perpendicular to the X″ axis isthe extension direction of the electrode fingers, as in the embodiments,it is possible to realize a low-loss SAW resonator, while maintaininggood temperature characteristics in the invention.

Here, a description will be made of the relationship between a stop bandupper end mode SAW and a lower end mode SAW. FIG. 4 is a diagramillustrating the distribution of the stop band upper end mode and lowerend mode standing waves in the normal IDT 12. In the stop band lower endmode and upper end mode SAWs formed by the kind of normal IDT 12 shownin FIG. 4 (what are shown in FIG. 4 are the electrode fingers 18configuring the IDT 12), the positions of the anti-nodes (or nodes) ofeach standing wave are separated from each other by π/2 (that is, λ/4).

According to FIG. 4, as described above, the anti-nodes of the stop bandlower end mode standing wave shown by the solid line exist in thecentral position of the electrode fingers 18, that is, in the reflectioncenter position, and the nodes of the stop band upper end mode standingwave shown by the dashed-dotted line exist in the reflection centerposition. In this kind of mode in which the nodes exist in the centralpositions between the electrode fingers, it is not possible toefficiently convert the oscillation of the SAW to a charge with theelectrode fingers 18 (18 a and 18 b), and it is often the case that itis not possible to excite or receive the mode as an electrical signal.However, with the method described in the application, by making theEuler angle Ψ other than zero, and displacing the SAW propagationdirection from the X axis, which is the crystal axis of the quartzcrystal, it is possible to shift the standing wave of the stop bandupper end mode to the position of the solid line in FIG. 4, that is, toshift the standing wave anti-nodes of the mode to the central positionof the electrode fingers 18, and it is possible to excite the SAW of thestop band upper end mode.

In addition, one pair of the reflectors 20 is provided so as to sandwichthe IDT 12 in the SAW propagation direction. As a specific configurationexample, both ends of each of a plurality of conductor strips 22,provided parallel to the electrode fingers 18 configuring the IDT 12,are connected.

Here, in another embodiment, only one set of ends of a plurality ofconductor strips 22 may be connected to each other. Further, in stillanother embodiment, parts (for example, centers in the extensiondirection of a plurality of conductor strips 22) other than both ends ofa plurality of conductor strips 22 may be connected to each other.

With an edge reflection type SAW resonator which actively utilizes areflected wave from the SAW propagation direction end face of the quartzcrystal substrate, or a multi-pair IDT type SAW resonator which excitesthe SAW standing wave with the IDT itself by increasing the number ofpairs of IDT electrode fingers, the reflectors are not absolutelynecessary.

As the material of the electrode film configuring the IDT 12 andreflectors 20 configured in this way, it is possible to use aluminum(Al), or a metal alloy with Al as its base.

By making the electrode thickness of the electrode film configuring theIDT 12 and reflectors 20 extremely small, the effect of the temperaturecharacteristics possessed by the electrodes is kept to a minimum.Furthermore, by making the depth of the quartz crystal substrate portiongrooves large, good frequency-temperature characteristics are derivedfrom the performance of the quartz crystal substrate portion grooves,that is, by utilizing the good temperature characteristics of the quartzcrystal. Because of this, it is possible to reduce the effect of theelectrode temperature characteristics on the temperature characteristicsof the SAW resonator and, provided that the fluctuation of the electrodemass is within 10%, it is possible to maintain good temperaturecharacteristics.

For the above-mentioned reasons, in a case of using an alloy as theelectrode film material, the proportion by weight of metals other thanthe main element Al should be 10% or less, and preferably 3% or less.With this, a case of using pure Al and a case of using an Al alloy canobtain good temperature characteristics or other electricalcharacteristics equivalent to each other.

In a case of using electrodes with a metal other than Al as a base, theelectrode film thickness may be adjusted so that the mass of theelectrodes is within ±10% of that when using Al. In this way, goodtemperature characteristics equivalent to those when using Al can beobtained.

The quartz crystal substrate 30 in the SAW resonator 10 with theabove-described basic configuration is provided with the grooves(inter-electrode finger grooves) 32 between the electrode fingers of theIDT 12 and between the conductor strips of the reflectors 20.

When the SAW wavelength in the stop band upper end mode is λ, and thegroove depth is G, the groves 32 provided in the quartz crystalsubstrate 30 may be such that0.01λ≦G  (1).

When fixing an upper limit for the groove depth G, as can be seen fromFIG. 5, it may be within the range of0.01λ≦G≦0.094λ  (2).

This is because, by fixing the groove depth G within this range, it ispossible to keep the frequency fluctuation amount within the operatingtemperature range (−40° C. to +85° C.) at or below the target value of25 ppm, to be described in detail later. In addition, it is preferablethat the groove depth G be within the range of0.01λ≦G≦0.0695λ  (3).

By fixing the groove depth G within this kind of range, even in a casewhere manufacturing variation occurs in the groove depth G, it ispossible to keep the shift amount of resonance frequency betweenindividual SAW resonators 10 within a correctable range.

In addition, the line occupation rate η is a value where the line widthL of the electrode finger 18 (in a case of the quartz crystal protrusiononly, the width of the protrusion) is divided by the pitch λ/2 (=L+S)between the electrode fingers 18, as shown in FIGS. 1C and 1D.Consequently, the line occupation rate η can be expressed by Equation(4).η=L/(L+S)  (4)

Here, in relation to the SAW resonator 10 according to the embodiment,the line occupation rate η may be fixed within a range which satisfiesExpressions (5) and (6). As can also be seen from Expressions (5) and(6), η can be derived by fixing the depth G of the grooves 32.−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775 provided that0.0100λ≦G≦0.0500λ  (5)−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775 provided that0.0500λ<G≦0.0695λ  (6)

In addition, it is preferable that the film thickness of the electrodefilm material (IDT 12, reflectors 20, and the like) in the SAW resonator10 according to the embodiment be within the range of0<H≦0.035λ  (7).

Furthermore, when taking into consideration the thickness of theelectrode film shown in Equation (7) with regard to the line occupationrate η, η can be obtained from Equation (8)

$\begin{matrix}{\eta = {{{- 1963.05} \times \left( {G/\lambda} \right)^{3}} + {196.28 \times \left( {G/\lambda} \right)^{2}} - {6.53 \times \left( {G/\lambda} \right)} - {135.99 \times \left( {H/\lambda} \right)^{2}} + {5.817 \times \left( {H/\lambda} \right)} + 0.732 - {99.99 \times \left( {G/\lambda} \right) \times \left( {H/\lambda} \right)}}} & (8)\end{matrix}$

With the manufacturing variation of the electrical characteristics(particularly the resonance frequency) being greater the greater theelectrode film thickness, there is a high probability that themanufacturing variation of the line occupation rate η may be within±0.04 or less when the electrode film thickness H is within the range ofExpressions (5) and (6), and that a manufacturing variation greater than±0.04 will occur when H>0.035λ. However, provided that the electrodefilm thickness H is within the range of Equations (5) and (6), and thevariation of the line occupation rate η is within ±0.04 or less, it ispossible to realize a SAW device with a low absolute value ofsecond-order temperature coefficient β. That is, a line occupation rateη up to the range of Equation (9), where a tolerance of ±0.04 is addedto Equation (8), is allowable.

$\begin{matrix}{\eta = {{{- 1963.05} \times \left( {G/\lambda} \right)^{3}} + {196.28 \times \left( {G/\lambda} \right)^{2}} - {6.53 \times \left( {G/\lambda} \right)} - {135.99 \times \left( {H/\lambda} \right)^{2}} + {5.817 \times \left( {H/\lambda} \right)} + 0.732 - {{99.99 \times \left( {G/\lambda} \right) \times \left( {H/\lambda} \right)} \pm 0.04}}} & (9)\end{matrix}$

In the SAW resonator 10 according to the embodiment with theabove-described kind of configuration, in a case where the second-ordertemperature coefficient β is within ±0.01 (ppm/° C.²), and the SAWoperating temperature range is preferably −40° C. to +85° C., the objectis to improve the frequency-temperature characteristics to a degreewhereby it is possible to keep a frequency fluctuation amount ΔF in theoperating temperature range at or under 25 ppm.

Generally, however, the temperature characteristics of a surfaceacoustic wave resonator are expressed by the following equation.Δf=α×(T−T ₀)+β×(T−T ₀)²

Here, Δf represents the frequency change amount (ppm) between thetemperature T and the peak temperature T₀, α represents the first-ordertemperature coefficient (ppm/° C.), β represents the second-ordertemperature coefficient (ppm/° C.²), T represents the temperature, andT₀ represents the temperature at which the frequency is highest (thepeak temperature).

For example, a case where the piezoelectric substrate is formed of aso-called ST cut (Euler angles (φ, θ, and Ψ)=(0°, 120° to 130°, and 0°)quartz crystal substrate, the first-order temperature coefficient α=0.0(ppm/° C.), and the second-order temperature coefficient β=−0.034 (ppm/°C.²), is expressed in a graph is as in FIG. 6. In FIG. 6, thetemperature characteristics describe an upwardly convex parabola(quadratic curve).

In the SAW resonator shown in FIG. 6, the frequency fluctuation amountwith respect to the temperature change is extremely large, and it isnecessary to suppress the frequency change amount Δf with respect to thetemperature change. Consequently, it is necessary to realize a surfaceacoustic wave resonator based on new findings, so that the second-ordertemperature coefficient β shown in FIG. 6 is brought closer to 0, andthe frequency change amount Δf with respect to the temperature(operating temperature) change when the SAW resonator is actually usedcomes closer to 0.

Consequently, an advantage of some aspects of the invention is to solvethe above-described kind of problem, making the frequency-temperaturecharacteristics of the surface acoustic wave device extremely good ones,and realizing a surface acoustic wave device which operates with astable frequency, even though the temperature changes.

How a solution to the above-described kind of problem may be realizedwith a SAW device to be configured including the above-described kind oftechnical sprit (technical components), that is, how the inventorarrived at the findings according to the invention by repeatedlycarrying out simulations and experiments, will be described in detailand proved hereinafter.

In a SAW resonator using the above-described quartz crystal substratecalled an ST cut, with the propagation direction as the crystal X axisdirection, in a case where the operating temperature range is the same,the frequency fluctuation amount ΔF in the operating temperature rangeis approximately 133 (ppm), and the second-order temperature coefficientβ is about −0.034 (ppm/° C.²). In addition, in a case of utilizing astop band lower end mode excitation in a SAW resonator using an in-planerotation ST cut quartz crystal substrate with the same operatingtemperature range, with the quartz crystal substrate cut angles and SAWpropagation direction as (0, 123°, and 45°) in Euler anglerepresentation, the frequency fluctuation amount ΔF is approximately 63ppm, and the second-order temperature coefficient β is about −0.016(ppm/° C.²).

The SAW resonators using the ST cut quartz crystal substrate andin-plane rotation ST cut quartz crystal substrate both utilizing surfaceacoustic waves called Rayleigh waves, have extremely small variation offrequency and frequency-temperature characteristics with respect to themachining accuracy of the quartz crystal substrate and electrodes incomparison with the surface acoustic wave, called a leaky wave, of anLST cut quartz crystal substrate, meaning that they are most suitablefor mass production, and are used in various kinds of SAW device.However, with the SAW resonators using the ST cut quartz crystalsubstrate, in-plane rotation ST cut quartz crystal substrate, or thelike utilized to date, as described above, the second-order temperaturecharacteristics being such that the curve indicating thefrequency-temperature characteristics is a quadratic curve, andfurthermore, the absolute value of the second-order temperaturecoefficient of the second-order temperature characteristics being large,the frequency fluctuation amount in the operating temperature range islarge, and they may not be utilized in a SAW device such as a resonatoror oscillator used in a wired communication device or wirelesscommunication device which requires a stability of frequency. Forexample, in a case where it is possible to obtain frequency-temperaturecharacteristics which have second-order temperature characteristicswhere the second-order temperature coefficient β is ±0.01 (ppm/° C.²) orless, corresponding to an improvement in the ST cut quartz crystalsubstrate second-order temperature coefficient β of ⅓ or less, and inthe in-plane rotation ST cut quartz crystal substrate second-ordertemperature coefficient β of 37% or more, it is possible to realize adevice which requires that kind of frequency stability. Furthermore, ina case where it is possible to obtain third-order temperaturecharacteristics, where the second-order temperature coefficient β issubstantially zero, and the curve indicating the frequency-temperaturecharacteristics is a cubic curve, it is more preferable, as thefrequency stability in the operating temperature range furtherincreases. In third-order temperature characteristics such as these, itis possible to obtain an extremely high frequency stability of ±25 ppmor less, which has not been realizable with the SAW device in therelated art, even in the wide operating temperature range of −40° C. to+85° C.

The fact that the electrode finger 18 line occupation rate η in the IDT12, electrode film thickness H, groove depth G, and the like, arerelated to the change in the frequency-temperature characteristics ofthe SAW resonator 10, as described above, has been made clear by thefindings based on the simulations and experiments carried out by theinventor. Then, the SAW resonator 10 according to the embodimentutilizes the excitation of the stop band upper end mode.

FIGS. 7A to 7D are graphs illustrating the change in the second-ordertemperature coefficient β with respect to the change in the lineoccupation rate η in a case of exciting and propagating a SAW on thesurface of the quartz crystal substrate 30, with the electrode filmthickness H in FIG. 1C as zero (H=0% λ), that is, in a condition inwhich the grooves 32 configured of uneven quartz crystal are formed inthe surface of the quartz crystal substrate 30. In FIGS. 7A to 7D, FIG.7A shows the second-order temperature coefficient β for the stop bandupper end mode resonance when the groove depth G is 0.02λ, and FIG. 7Bshows the second-order temperature coefficient β for the stop band lowerend mode resonance when the groove depth G is 0.02λ. In addition, ofFIGS. 7A to 7D, FIG. 7C shows the second-order temperature coefficient βfor the stop band upper end mode resonance when the groove depth G is0.04λ, and FIG. 7D shows the second-order temperature coefficient β forthe stop band lower end mode resonance when the groove depth G is 0.04λ.The simulations shown in FIGS. 7A to 7D show examples of cases in whicha SAW is propagated in some way in a quartz crystal substrate 30 onwhich no electrode film is provided, in order to reduce factors causingthe frequency-temperature characteristics to fluctuate. In addition,Euler angles (0°, 123°, and Ψ) are used for the cut angles of the quartzcrystal substrate 30. With regard to Ψ, a value which is the minimumabsolute value of the second-order temperature coefficient β is selectedas appropriate.

From FIGS. 7A to 7D, it can be seen that, both in the case of the stopband upper end mode and in the case of the lower end mode, thesecond-order temperature coefficient β changes considerably in the areain which the line occupation rate η is 0.6 to 0.7. Then, when comparingthe change in the second-order temperature coefficient β in the stopband upper end mode and the change in the second-order temperaturecoefficient β in the stop band lower end mode, the following points canbe seen. That is, the change in the second-order temperature coefficientβ in the stop band lower end mode is such that the characteristicsdeteriorate due to the second-order temperature coefficient β changingfrom the minus side to farther on the minus side (the absolute value ofthe second-order temperature coefficient β increases). As opposed tothis, the change in the second-order temperature coefficient β in thestop band upper end mode is such that the characteristics improve due tothe second-order temperature coefficient β changing from the minus sideto the plus side (a point exists at which the absolute value of thesecond-order temperature coefficient β decreases).

From this, it is clear that in order to obtain goodfrequency-temperature characteristics in a SAW device, it is preferableto use the oscillation of the stop band upper end mode.

Next, the inventor investigated the relationship between the lineoccupation rate η and second-order temperature coefficient β whenpropagating a stop band upper end mode SAW in quartz crystal substrateswith variously changed groove depths G.

FIGS. 8A to 8I are graphs illustrating evaluation results whensimulating the relationship between the line occupation rate η andsecond-order temperature coefficient β when changing the groove depth Gfrom 0.01λ (1% λ) to 0.08λ (8% λ), with the electrode film thickness Has zero (H=0% λ), as in FIGS. 7A to 7D. From the evaluation results, itcan be seen that a point at which β=0, that is, a point at which theapproximate curve indicating the frequency-temperature characteristicsdescribes a cubic curve, begins to appear from around the point at whichthe groove depth G is made to be 0.0125λ (1.25% λ), as shown in FIG. 8B.Then, it is also clear from FIGS. 8A to 8I that η at which β=0 exists intwo places (a point (η1) at which β=0 where η is larger, and a point(η2) at which β=0 where η is smaller). Furthermore, it can also be seenfrom the evaluation results shown in FIGS. 8A to 8I that the fluctuationamount of the line occupation rate η with respect to the change in thegroove depth G is greater at η2 than at η1.

Regarding this, it is possible to increase an understanding thereof byreferring to FIG. 9. FIG. 9 is a graph plotting η1 and η2, at which thesecond-order temperature coefficient β becomes 0, in a case of changingthe groove depth G. From FIG. 9, it can be seen that η1 and η2 bothbecome smaller as the groove depth G increases, but the fluctuationamount of η2 is so great that, on a graph in which the scale of thevertical axis η is shown in a range of 0.5λ to 0.9λ, it goes off thescale around the point at which the groove depth G=0.04λ. That is tosay, it can be said that the fluctuation amount of η2 with respect tothe change in the groove depth G is large.

FIG. 10A to FIG. 10I are graphs with the electrode film thickness H aszero (H=0%λ), as in FIGS. 7A to 8I, and with the vertical axis of FIGS.8A to 8I shown as the frequency fluctuation amount ΔF instead of thesecond-order temperature coefficient β. From FIGS. 10A to 10I, it can ofcourse be seen that the frequency fluctuation amount ΔF decreases at thetwo points (η1 and η12) at which β=0. Furthermore, it can be seen fromFIGS. 10A to 10I that of the two points at which β=0, the frequencyfluctuation amount ΔF is kept smaller at the point corresponding to η1in every graph in which the groove depth G is changed.

According to the above-described tendency, it can be supposed that it ispreferable to employ the β=0 point at which the frequency fluctuationamount with respect to the fluctuation in the groove depth G is smaller,that is, η1, for a mass production article in which discrepancies areliable to occur when manufacturing. FIG. 5 shows a graph of therelationship between the frequency fluctuation amount ΔF at the point(η1) at which the second-order temperature coefficient β is smallest andthe groove depth G, for each groove depth G. According to FIG. 5, withthe lower limit value of the groove depth G for which the frequencyfluctuation amount ΔF is the target value of 25 ppm or less being thegroove depth G of 0.01λ, the range of the groove depth G is that orgreater, that is, 0.01≦G.

In FIG. 5, examples are also added of cases in which, by simulation, thegroove depth G is 0.08 or more. According to the simulation, thefrequency fluctuation amount ΔF is 25 ppm or less when the groove depthG is 0.01λ or more, and subsequently, the frequency fluctuation amountΔF decreases every time the groove depth G increases. However, when thegroove depth G becomes approximately 0.9λ or more, the frequencyfluctuation amount ΔF increases again, and on the groove depth Gexceeding 0.094λ, the frequency fluctuation amount ΔF exceeds 25 ppm.

The graph shown in FIG. 5 is of a simulation in a condition in which noelectrode film is formed on the IDT 12, reflectors 20, and the like, onthe quartz crystal substrate 30 but, as can be understood by referringto FIGS. 21A to 26F, whose details are shown hereinafter, it is supposedthat it is possible to reduce the frequency fluctuation amount ΔF morein a SAW resonator 10 on which an electrode film is provided. Therefore,when fixing the upper limit value of the groove depth G, it may be madethe maximum value in the condition in which no electrode film is formed,that is, G≦0.94λ, and it is possible to represent the preferred range ofthe groove depth G for achieving the target as0.01λ≦G≦0.094λ  (2).

The groove depth G has a maximum variation of around ±0.001λ in the massproduction process. Therefore, the individual frequency fluctuationamounts Δf of the SAW resonator 10 in a case in which the groove depth Gdeviates by ±0.001λ, when the line occupation rate η is taken to be aconstant, are shown in FIG. 11. According to FIG. 11, it can be seenthat in a case in which the groove depth G deviates by ±0.001λ whenG=0.04λ, that is, when the groove depth is in a range of0.039λ≦G≦0.041λ, the frequency fluctuation amount ΔF is around ±500 ppm.

Here, in a case where the frequency fluctuation amount ΔF is less than±1000 ppm, frequency adjustment is possible using various frequency fineadjustment unit. However, in a case where the frequency fluctuationamount Δf is ±1000 ppm or more, adjusting the frequency has an effect onstatic characteristics such as the Q value and CI (crystal impedance)value, and on long-term reliability, leading to a reduction in the yieldrate as the SAW resonator 10.

By deriving an approximate equation indicating the relationship betweenthe frequency fluctuation amount Δf (ppm) and groove depth G for thestraight line linking the plots shown in FIG. 11, it is possible toobtain Equation (10).Δf=16334(G/λ)−137  (10)

Here, on calculating the values of G at which Δf<1000 ppm, it is foundthat G≦0.0695%. Consequently, it can be said that it is preferable thatthe range of the groove depth G according to the embodiment be optimally0.01λ≦G≦0.0695λ  (3).

Next, FIGS. 12A and 12F are graphs illustrating evaluation results whensimulating the relationship between η at which the second-ordertemperature coefficient β=0, that is, the line occupation rate ηindicating the third-order temperature characteristics, and the groovedepth G. The quartz crystal substrate 30 has Euler angles of (0°, 123°,and Ψ). Here, Ψ is appropriately selected as the angle at which thefrequency-temperature characteristics indicate the tendency of the cubiccurve, that is, the angle at which the second-order temperaturecoefficient β=0. The relationship between the Euler φ angle and thegroove depth G when obtaining η at which β=0, under the same conditionsas in FIGS. 12A to 12F, is shown in FIGS. 34A to 34F. Of FIGS. 34A to34F, in the graph (FIG. 34C) in which the electrode film thicknessH=0.02λ, although the plots of Ψ<42° are not shown, the 12 plot of thegraph is Ψ=41.9° at G=0.03λ. The plots of the relationship between thegroove depth G and line occupation rate η for each electrode filmthickness are obtained based on FIG. 15A to FIG. 20F, which will bedescribed in detail hereinafter.

From the evaluation results shown in FIGS. 12A to 12F, it can be seenthat for every film thickness, as described above, the fluctuation of η1due to the change in the groove depth G is small in comparison with thatof η2. For this reason, η1 has been extracted from the graphsillustrating the relationship between the groove depth G and lineoccupation rate η for each film thickness in FIGS. 12A to 12F, andsummarized by plotting points at which β≅0 in FIG. 13A. In contrast, onevaluating an area in which, although not reaching β≅0, |β|≦0.01 (ppm/°C.²) is satisfied, it becomes clear that η1 are concentrated in thepolygon indicated by the solid line, as shown in FIG. 13B.

The coordinates of points a to h in FIG. 13B are shown in Table 1 below.

TABLE 1 Point G/λ η a 0.01 0.70 b 0.03 0.66 c 0.05 0.62 d 0.07 0.55 e0.07 0.60 f 0.05 0.65 g 0.03 0.70 h 0.01 0.75

FIG. 13B shows that, provided that it is inside the polygon surroundedby points a to h, |β|≦0.01 (ppm/° C.²) is guaranteed regardless of thesize of the electrode film thickness H, and good frequency-temperaturecharacteristics can be obtained. A range within which the goodfrequency-temperature characteristics can be obtained is a range whichsatisfies both Equations (11) and (12), and Equation (13), shown below.η≦−2.5000×G/λ+0.7775 provided that 0.0100λ≦G≦0.0695λ  (11)η≦−2.0000×G/λ+0.7200 provided that 0.0100λ≦G≦0.0500λ  (12)η3.5898×G/λ+0.7995 provided that 0.0500λ<G≦0.0695λ  (13)

From Equations (11), (12), and (13), it can be said that it is possibleto specify the line occupation rates η in the range surrounded by thesolid line in FIG. 13B as a range satisfying both Equation (5) andEquation (6).−2.0000×G/λ+0.7200≦η≦−−2.5000×G/λ+0.7775 provided that0.0100λ≦G≦0.0500λ  (5)−3.5898×G/λ+0.7995≦η≦−−2.5000×G/λ+0.7775 provided that0.0500λ<G≦0.0695λ  (6)

Here, in a case of allowing the second-order temperature coefficient βto be within ±0.01 (ppm/° C.²), it is confirmed that by configuring soas to satisfy both Equation (3) and Equation (5) when 0.0100λ≦G≦0.0500λ,and satisfy both Equation (3) and Equation (6) when 0.0500λ≦G≦0.0695λ,the second-order temperature coefficient β comes within ±0.01 (ppm/°C.²).

The values of the second-order temperature coefficient β (ppm/° C.²) foreach electrode film thickness H at the points a to h are shown in Table2 below. From Table 2, it can be confirmed that |β|≦0.01 (ppm/° C.²) atall of the points.

TABLE 2 Electrode Film Thickness H Point 1% λ 1.5% λ 2% λ 2.5% λ 3% λ3.5% λ a −0.099 × 10⁻¹ −0.070 × 10⁻¹ −0.030 × 10⁻¹  0.030 × 10⁻¹ −0.050× 10⁻¹ −0.060 × 10⁻¹ b  0.040 × 10⁻¹  0.030 × 10⁻¹  0.000 × 10⁻¹  0.000× 10⁻¹ −0.020 × 10⁻¹ −0.040 × 10⁻¹ c  0.070 × 10⁻¹ −0.040 × 10⁻¹  0.010× 10⁻¹ −0.036 × 10⁻¹ −0.040 × 10⁻¹ −0.057 × 10⁻¹ d  0.067 × 10⁻¹ −0.022× 10⁻¹ −0.070 × 10⁻¹ −0.080 × 10⁻¹ −0.090 × 10⁻¹ −0.099 × 10⁻¹ e −0.039× 10⁻¹ −0.060 × 10⁻¹ −0.090 × 10⁻¹ −0.080 × 10⁻¹ −0.090 × 10⁻¹ −0.094 ×10⁻¹ f −0.023 × 10⁻¹ −0.070 × 10⁻¹ −0.050 × 10⁻¹ −0.062 × 10⁻¹ −0.060 ×10⁻¹ −0.070 × 10⁻¹ g −0.070 × 10⁻¹ −0.060 × 10⁻¹ −0.090 × 10⁻¹ −0.070 ×10⁻¹ −0.070 × 10⁻¹ −0.070 × 10⁻¹ h −0.099 × 10⁻¹ −0.030 × 10⁻¹ −0.091 ×10⁻¹ −0.080 × 10⁻¹ −0.080 × 10⁻¹ −0.080 × 10⁻¹

In addition, when the relationship between the groove depth G and lineoccupation rate η at each point at which β=0 for SAW resonators 10 inwhich the electrode film thickness H≅0, 0.01λ, 0.02λ, 0.03λ, based onEquations (11) to (13) and Equations (5) and (6) derived therefrom, isindicated by an approximate line, the result is as in FIG. 14. Therelationship between the groove depth G and line occupation rate η inthe quartz crystal substrate 30 on which no electrode film is providedis as shown in FIG. 9.

When changing the electrode film thickness H at 3.0% λ(0.030λ) or less,the frequency-temperature characteristics of β=0, that is, the cubiccurve, can be obtained. At this time, a relational equation for G and ηwhen the frequency-temperature characteristics are good can be expressedby Equation (8).

$\begin{matrix}{\eta = {{{- 1963.05} \times \left( {G/\lambda} \right)^{3}} + {196.28 \times \left( {G/\lambda} \right)^{2}} - {6.53 \times \left( {G/\lambda} \right)} - {135.99 \times \left( {H/\lambda} \right)^{2}} + {5.817 \times \left( {H/\lambda} \right)} + 0.732 - {99.99 \times \left( {G/\lambda} \right) \times \left( {H/\lambda} \right)}}} & (8)\end{matrix}$

Here, the units of G and H are λ.

It is noted that Equation (8) is established when the electrode filmthickness H is in the range of 0<H≦0.030λ.

With the manufacturing variation of the electrical characteristics(particularly the resonance frequency) being greater the greater theelectrode film thickness, there is a high probability that themanufacturing variation of the line occupation rate η may be ±0.04 orless when the electrode film thickness H is within the range ofEquations (5) and (6), and that a manufacturing variation greater than±0.04 will occur when H>0.035×. However, provided that the electrodefilm thickness H is within the range of Equations (5) and (6), and thevariation of the line occupation rate η is ±0.04 or less, it is possibleto realize a SAW device with a low second-order temperature coefficientβ. That is, when taking into consideration the manufacturing variationof the line occupation rate, and keeping the second-order temperaturecoefficient β within ±0.01 ppm/° C.², a line occupation rate η up to therange of Equation (9), where a tolerance of ±0.04 is added to Equation(8), is allowable.

$\begin{matrix}{\eta = {{{- 1963.05} \times \left( {G/\lambda} \right)^{3}} + {196.28 \times \left( {G/\lambda} \right)^{2}} - {6.53 \times \left( {G/\lambda} \right)} - {135.99 \times \left( {H/\lambda} \right)^{2}} + {5.817 \times \left( {H/\lambda} \right)} + 0.732 - {{99.99 \times \left( {G/\lambda} \right) \times \left( {H/\lambda} \right)} \pm 0.04}}} & (9)\end{matrix}$

FIGS. 15A to 20F show graphs of the relationship between the lineoccupation rate η and second-order temperature coefficient β when thegroove depth G is changed, in cases in which the electrode filmthickness is 0.01λ(1% λ), 0.015λ(1.5% λ), 0.02λ(2% λ), 0.025λ(2.5% λ),0.03λ(3% λ), and 0.035λ(3.5% λ), respectively.

In addition, FIGS. 21A to 26F show graphs of the relationship betweenthe line occupation rate η and frequency fluctuation amount ΔF in theSAW resonator 10 corresponding to each of the FIGS. 15A to 20F. Thequartz crystal substrates used are all ones with Euler angles (0°, 123°,and Ψ), and with regard to Ψ, an angle at which ΔF is smallest isappropriately selected.

Here, FIGS. 15A to 15F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.01λ, and FIGS.21A to 21F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.01λ.

In addition, FIGS. 16A to 16F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.015λ, and FIGS.22A to 22F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.015λ.

In addition, FIGS. 17A to 17F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.02λ, and FIGS.23A to 23F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.02λ.

In addition, FIGS. 18A to 18F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.025λ, and FIGS.24A to 24F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.025λ.

In addition, FIGS. 19A to 19F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.03λ, and FIGS.25A to 25F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.03λ.

In addition, FIGS. 20A to 20F are diagrams illustrating the relationshipbetween the line occupation rate η and second-order temperaturecoefficient β when the electrode film thickness H is 0.035λ, and FIGS.26A to 26F are diagrams illustrating the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when the electrodefilm thickness H is 0.035λ.

Although there are slight differences in all of the graphs in thesediagrams (FIGS. 15A to 26F), regarding the change tendencies thereof, itis seen that they are similar to those in FIGS. 8A to 8I and 10A to 10I,which are graphs illustrating the relationship between the lineoccupation rate η and second-order temperature coefficient β, and lineoccupation rate η and frequency fluctuation amount ΔF, in the quartzcrystal substrate 30 only.

That is, it can be said that an advantage according to the embodiment ofthe invention is that it can be achieved even when propagating a surfaceacoustic wave on an individual quartz crystal substrate 30 from whichthe electrode film is omitted.

For each of the two points η1 and η2 at which the second-ordertemperature coefficient β becomes zero, a simulation is performed foreach of the range of η1 and η2 when the range of β is expanded to|β|≦0.01 (ppm/° C.²), and the case in which the range of the electrodefilm thickness H is fixed, and the groove depth G is changed. Of η1 andη2, the larger η at which |β|≦0.01 (ppm/° C.²) is taken to be η1, andthe smaller at which |β|≦0.01 (ppm/° C.²) is η2. The quartz crystalsubstrates used are all ones with Euler angles (0°, 123°, and Ψ), andwith regard to Ψ, an angle at which ΔF is smallest is appropriatelyselected.

FIG. 27A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.000λ<H≦0.005λ, and Table 3 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 27A, and the value of β at themeasurement points.

TABLE 3 Point G/λ η β (ppm/° C.²) a 0.010 0.710 −0.098 × 10⁻¹ b 0.0200.710 −0.099 × 10⁻¹ c 0.030 0.710 −0.095 × 10⁻¹ d 0.040 0.710 −0.100 ×10⁻¹ e 0.050 0.710 −0.100 × 10⁻¹ f 0.060 0.710 −0.098 × 10⁻¹ g 0.0700.710 −0.099 × 10⁻¹ h 0.080 0.710 −0.097 × 10⁻¹ i 0.090 0.710 −0.100 ×10⁻¹ j 0.090 0.420  0.073 × 10⁻¹ k 0.080 0.570  0.086 × 10⁻¹ l 0.0700.590  0.093 × 10⁻¹ m 0.060 0.615  0.077 × 10⁻¹ n 0.050 0.630  0.054 ×10⁻¹ o 0.040 0.635  0.097 × 10⁻¹ p 0.030 0.650  0.097 × 10⁻¹ q 0.0200.670  0.074 × 10⁻¹ r 0.010 0.710  0.091 × 10⁻¹

From FIG. 27A and Table 3, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range 0.01λ≦G≦0.09λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to r as the vertices.

FIG. 27B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.000λ<H≦0.005λ, and Table 4 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 27B, and the value of β at themeasurement points.

TABLE 4 Point G/λ η β (ppm/° C.²) a 0.030 0.590 0.097 × 10⁻¹ b 0.0400.580 0.097 × 10⁻¹ c 0.050 0.550 0.054 × 10⁻¹ d 0.060 0.520 0.077 × 10⁻¹e 0.070 0.480 0.093 × 10⁻¹ f 0.080 0.450 0.086 × 10⁻¹ g 0.090 0.4000.073 × 10⁻¹ h 0.090 0.180 0.056 × 10⁻¹ i 0.080 0.340 0.093 × 10⁻¹ j0.070 0.410 0.078 × 10⁻¹ k 0.060 0.460 0.094 × 10⁻¹ l 0.050 0.490 0.085× 10⁻¹ m 0.040 0.520 0.099 × 10⁻¹ n 0.030 0.550 0.098 × 10⁻¹

From FIG. 27B and Table 4, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.03λ≦G≦0.09λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 28A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.005λ<H≦0.010λ, and Table 5 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 28A, and the value of β at themeasurement points.

TABLE 5 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.099 × 10⁻¹   b 0.0200.740 −0.100 × 10⁻¹   c 0.030 0.715 −0.100 × 10⁻¹   d 0.040 0.730 −0.098× 10⁻¹   e 0.050 0.740 −0.100 × 10⁻¹   f 0.060 0.730 −0.098 × 10⁻¹   g0.070 0.730 −0.100 × 10⁻¹   h 0.080 0.730 −0.100 × 10⁻¹   i 0.080 0.5000.086 × 10⁻¹ j 0.070 0.570 0.100 × 10⁻¹ k 0.060 0.610 0.095 × 10⁻¹ l0.050 0.630 0.100 × 10⁻¹ m 0.040 0.635 0.097 × 10⁻¹ n 0.030 0.655 0.070× 10⁻¹ o 0.020 0.680 0.100 × 10⁻¹ p 0.010 0.760 0.016 × 10⁻¹

From FIG. 28A and Table 5, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.08λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to p as the vertices.

FIG. 28B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.005λ<H≦0.010λ, and Table 6 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 28B, and the value of β at themeasurement points.

TABLE 6 Point G/λ η β (ppm/° C.²) a 0.020 0.650   0.090 × 10⁻¹ b 0.0300.610   0.098 × 10⁻¹ c 0.040 0.570   0.097 × 10⁻¹ d 0.050 0.550   0.040× 10⁻¹ e 0.060 0.520   0.066 × 10⁻¹ f 0.070 0.470   0.070 × 10⁻¹ g 0.0700.370 −0.094 × 10⁻¹ h 0.060 0.440 −0.096 × 10⁻¹ i 0.050 0.480 −0.096 ×10⁻¹ j 0.040 0.520 −0.095 × 10⁻¹ k 0.030 0.550 −0.099 × 10⁻¹ l 0.0200.590 −0.100 × 10⁻¹

From FIG. 28B and Table 6, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.02λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to l as the vertices.

FIG. 29A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.010λ<H≦0.015λ, and Table 7 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 29A, and the value of β at themeasurement points.

TABLE 7 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.099 × 10⁻¹ b 0.0200.760 −0.099 × 10⁻¹ c 0.030 0.760 −0.099 × 10⁻¹ d 0.040 0.750 −0.099 ×10⁻¹ e 0.050 0.750 −0.099 × 10⁻¹ f 0.060 0.750 −0.099 × 10⁻¹ g 0.0700.740 −0.099 × 10⁻¹ h 0.080 0.740 −0.098 × 10⁻¹ i 0.080 0.340   0.088 ×10⁻¹ j 0.070 0.545   0.088 × 10⁻¹ k 0.060 0.590   0.099 × 10⁻¹ l 0.0500.620   0.090 × 10⁻¹ m 0.040 0.645   0.060 × 10⁻¹ n 0.030 0.670   0.030× 10⁻¹ o 0.020 0.705   0.076 × 10⁻¹ p 0.010 0.760   0.010 × 10⁻¹

From FIG. 29A and Table 7, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.08λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to p as the vertices.

FIG. 29B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.010λ<H≦0.015μ, and Table 8 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 29B, and the value of β at themeasurement points.

TABLE 8 Point G/λ η β (ppm/° C.²) a 0.010 0.740   0.099 × 10⁻¹ b 0.0200.650   0.090 × 10⁻¹ c 0.030 0.610   0.090 × 10⁻¹ d 0.040 0.570   0.080× 10⁻¹ e 0.050 0.540   0.060 × 10⁻¹ f 0.060 0.480   0.060 × 10⁻¹ g 0.0700.430   0.099 × 10⁻¹ h 0.070 0.350 −0.099 × 10⁻¹ i 0.060 0.420 −0.090 ×10⁻¹ j 0.050 0.470 −0.090 × 10⁻¹ k 0.040 0.510 −0.090 × 10⁻¹ l 0.0300.550 −0.090 × 10⁻¹ m 0.020 0.610 −0.099 × 10⁻¹ n 0.010 0.700 −0.099 ×10⁻¹

From FIG. 29B and Table 8, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 30A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.015λ<H≦0.020μ, and Table 9 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 30A, and the value of β at themeasurement points.

TABLE 9 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.100 × 10⁻¹ b 0.0200.770 −0.100 × 10⁻¹ c 0.030 0.760 −0.100 × 10⁻¹ d 0.040 0.760 −0.100 ×10⁻¹ e 0.050 0.760 −0.100 × 10⁻¹ f 0.060 0.750 −0.100 × 10⁻¹ g 0.0700.750 −0.100 × 10⁻¹ h 0.070 0.510   0.100 × 10⁻¹ i 0.060 0.570   0.099 ×10⁻¹ j 0.050 0.620   0.097 × 10⁻¹ k 0.040 0.640   0.096 × 10⁻¹ l 0.0300.660   0.080 × 10⁻¹ m 0.020 0.670   0.076 × 10⁻¹ n 0.010 0.700   0.010× 10⁻¹

From FIG. 30A and Table 9, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 30B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.015λ<H≦0.020λ, and Table 10 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 30B, and the value of β at themeasurement points.

TABLE 10 Point G/λ η β (ppm/° C.²) a 0.010 0.690   0.010 × 10⁻¹ b 0.0200.640   0.090 × 10⁻¹ c 0.030 0.590   0.090 × 10⁻¹ d 0.040 0.550   0.080× 10⁻¹ e 0.050 0.510   0.080 × 10⁻¹ f 0.060 0.470   0.090 × 10⁻¹ g 0.0700.415   0.100 × 10⁻¹ h 0.070 0.280 −0.100 × 10⁻¹ i 0.060 0.380 −0.090 ×10⁻¹ j 0.050 0.470 −0.090 × 10⁻¹ k 0.040 0.510 −0.090 × 10⁻¹ l 0.0300.550 −0.090 × 10⁻¹ m 0.020 0.610 −0.100 × 10⁻¹ n 0.010 0.680 −0.100 ×10⁻¹

From FIG. 30B and Table 10, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 31A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.020λ<H≦0.025λ, and Table 11 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 31A, and the value of β at themeasurement points.

TABLE 11 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.100 × 10⁻¹ b 0.0200.770 −0.100 × 10⁻¹ c 0.030 0.760 −0.100 × 10⁻¹ d 0.040 0.760 −0.100 ×10⁻¹ e 0.050 0.760 −0.096 × 10⁻¹ f 0.060 0.760 −0.100 × 10⁻¹ g 0.0700.760 −0.100 × 10⁻¹ h 0.070 0.550   0.100 × 10⁻¹ i 0.060 0.545   0.090 ×10⁻¹ j 0.050 0.590   0.097 × 10⁻¹ k 0.040 0.620   0.100 × 10⁻¹ l 0.0300.645   0.100 × 10⁻¹ m 0.020 0.680   0.070 × 10⁻¹ n 0.010 0.700   0.030× 10⁻¹

From FIG. 31A and Table 11, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 31B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.020λ<H≦0.025λ, and Table 12 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 31B, and the value of β at themeasurement points.

TABLE 12 Point G/λ η β (ppm/° C.²) a 0.010 0.690   0.030 × 10⁻¹ b 0.0200.640   0.090 × 10⁻¹ c 0.030 0.590   0.090 × 10⁻¹ d 0.040 0.550   0.090× 10⁻¹ e 0.050 0.510   0.080 × 10⁻¹ f 0.060 0.420   0.090 × 10⁻¹ g 0.0700.415   0.080 × 10⁻¹ h 0.070 0.340 −0.098 × 10⁻¹ i 0.060 0.340 −0.100 ×10⁻¹ j 0.050 0.420 −0.100 × 10⁻¹ k 0.040 0.470 −0.100 × 10⁻¹ l 0.0300.520 −0.093 × 10⁻¹ m 0.020 0.580 −0.100 × 10⁻¹ n 0.010 0.650 −0.090 ×10⁻¹

From FIG. 31B and Table 12, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 32A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.025λ<H≦0.030λ, and Table 13 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 32A, and the value of β at themeasurement points.

TABLE 13 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.098 × 10⁻¹ b 0.0200.770 −0.100 × 10⁻¹ c 0.030 0.770 −0.100 × 10⁻¹ d 0.040 0.760 −0.100 ×10⁻¹ e 0.050 0.760 −0.099 × 10⁻¹ f 0.060 0.760 −0.100 × 10⁻¹ g 0.0700.760 −0.100 × 10⁻¹ h 0.070 0.550   0.080 × 10⁻¹ i 0.060 0.505   0.087 ×10⁻¹ j 0.050 0.590   0.090 × 10⁻¹ k 0.040 0.620   0.100 × 10⁻¹ l 0.0300.645   0.100 × 10⁻¹ m 0.020 0.680   0.083 × 10⁻¹ n 0.010 0.700   0.052× 10⁻¹

From FIG. 32A and Table 13, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 32B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.025λ<H≦0.030λ, and Table 14 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 32B, and the value of β at themeasurement points.

TABLE 14 Point G/λ η β (ppm/° C.²) a 0.010 0.670   0.052 × 10⁻¹ b 0.0200.605   0.081 × 10⁻¹ c 0.030 0.560   0.092 × 10⁻¹ d 0.040 0.520   0.099× 10⁻¹ e 0.050 0.470   0.086 × 10⁻¹ f 0.060 0.395   0.070 × 10⁻¹ g 0.0700.500   0.080 × 10⁻¹ h 0.070 0.490 −0.100 × 10⁻¹ i 0.060 0.270 −0.100 ×10⁻¹ j 0.050 0.410 −0.100 × 10⁻¹ k 0.040 0.470 −0.100 × 10⁻¹ l 0.0300.520 −0.093 × 10⁻¹ m 0.020 0.580 −0.099 × 10⁻¹ n 0.010 0.620 −0.090 ×10⁻¹

From FIG. 32B and Table 14, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 33A is a graph illustrating the relationship between η1 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.030λ<H≦0.035λ, and Table 15 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 33A, and the value of β at themeasurement points.

TABLE 15 Point G/λ η β (ppm/° C.²) a 0.010 0.770 −0.100 × 10⁻¹ b 0.0200.770 −0.098 × 10⁻¹ c 0.030 0.770 −0.100 × 10⁻¹ d 0.040 0.760 −0.100 ×10⁻¹ e 0.050 0.760 −0.100 × 10⁻¹ f 0.060 0.760 −0.100 × 10⁻¹ g 0.0700.760 −0.100 × 10⁻¹ h 0.070 0.550   0.090 × 10⁻¹ i 0.060 0.500   0.087 ×10⁻¹ j 0.050 0.545   0.090 × 10⁻¹ k 0.040 0.590   0.091 × 10⁻¹ l 0.0300.625   0.080 × 10⁻¹ m 0.020 0.650   0.083 × 10⁻¹ n 0.010 0.680   0.093× 10⁻¹

From FIG. 33A and Table 15, it can be seen that when the electrode filmthickness H at η1 is within the above-described range, and when thegroove depth G is in the range 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

FIG. 33B is a graph illustrating the relationship between η2 whichsatisfies the above-described range of β and the groove depth G, whenthe electrode film thickness H is 0.030λ<H≦0.035λ, and Table 16 is atable illustrating the coordinates (G/λ, η) of principal measurementpoints for fixing the range shown in FIG. 33B, and the value of β at themeasurement points.

TABLE 16 Point G/λ η β (ppm/° C.²) a 0.010 0.655   0.080 × 10⁻¹ b 0.0200.590   0.081 × 10⁻¹ c 0.030 0.540   0.092 × 10⁻¹ d 0.040 0.495   0.099× 10⁻¹ e 0.050 0.435   0.090 × 10⁻¹ f 0.060 0.395   0.061 × 10⁻¹ g 0.0700.500   0.090 × 10⁻¹ h 0.070 0.550 −0.100 × 10⁻¹ i 0.060 0.380 −0.090 ×10⁻¹ j 0.050 0.330 −0.100 × 10⁻¹ k 0.040 0.410 −0.095 × 10⁻¹ l 0.0300.470 −0.099 × 10⁻¹ m 0.020 0.520 −0.100 × 10⁻¹ n 0.010 0.590 −0.100 ×10⁻¹

From FIG. 33B and Table 16, it can be seen that when the electrode filmthickness H at η2 is within the above-described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theabove-described requirement in the area surrounded by the polygon withthe measurement points a to n as the vertices.

The relationship between Ψ and the groove depth G obtained from η1 inthe graphs shown in FIGS. 34A to 34F is summarized in FIG. 35. Thereason for selecting η1 is as described above. As shown in FIG. 35,there is hardly any change in the angle Ψ, even though the thickness ofthe electrode film changes, and it is seen that the optimum angle Ψchanges in accordance with the fluctuation of the groove depth G. It canbe said that this too is proof that a high proportion of the change inthe second-order temperature coefficient β is due to the form of thequartz crystal substrate 30.

In the same way as described above, the relationships between the groovedepth G and Ψ when the second-order temperature coefficient β=−0.01(ppm/° C.²), and Ψ when β=+0.01 (ppm/° C.²), are obtained, andsummarized in FIGS. 36 and 37. By obtaining from these graphs (FIGS. 35to 37) the angles Ψ at which it is possible to achieve −0.01≦β≦+0.01, itis possible to fix the preferable Ψ angle range under theabove-described conditions at 43°<Ψ<45°, and it is possible to morepreferably fix the range at 43.2°≦Ψ≦44.2°.

A simulation is carried out for the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when changing the groove depth G, inthe case of changing the electrode film thickness H. The results of thesimulation are shown in FIGS. 38A to 44B. The quartz crystal substratesused are all ones with Euler angles (0°, 123°, and Ψ), and with regardto Ψ, an angle at which ΔF is smallest is appropriately selected.

FIG. 38A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0<H≦0.005%. Here, the range sandwiched by the straightline connecting the plots indicating the maximum value of Ψ and thebroken line connecting the plots indicating the minimum value of Ψ isthe range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.38A as a polygon, it can be shown as in FIG. 38B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 38B.When expressing the range of the polygon shown in FIG. 38B in anapproximate equation, it can be expressed by Equations (14) and (15).Ψ≦3.0×G/λ+43.92 provided that 0.0100λ≦G≦0.0695λ  (14)Ψ≧−48.0×G/λ+44.35 provided that 0.0100λ≦G≦0.0695λ  (15)

FIG. 39A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.005λ≦H≦0.010λ. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.39A as a polygon, it can be shown as in FIG. 39B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 39B.When expressing the range of the polygon shown in FIG. 39B in anapproximate equation, it can be expressed by Equations (16) and (17).Ψ≦8.0×G/λ+43.60 provided that 0.0100λ≦G≦0.0695λ  (16)Ψ≧−48.0×G/λ+44.00 provided that 0.0100λ≦G≦0.0695λ  (17)

FIG. 40A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.010λ≦H≦0.015%. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.40A as a polygon, it can be shown as in FIG. 40B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 40B.When expressing the range of the polygon shown in FIG. 40B in anapproximate equation, it can be expressed by Equations (18) and (19).Ψ≦10.0×G/λ+43.40 provided that 0.0100λ≦G≦0.0695λ  (18)Ψ≧−44.0×G/λ+43.80 provided that 0.0100λ≦G≦0.0695λ  (19)

FIG. 41A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.015λ≦H≦0.020λ. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.41A as a polygon, it can be shown as in FIG. 41B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 41B.When expressing the range of the polygon shown in FIG. 41B in anapproximate equation, it can be expressed by Equations (20) and (21).Ψ≦12.0×G/λ+43.31 provided that 0.0100λ≦G≦0.0695λ  (20)Ψ≧−30.0×G/λ+43.40 provided that 0.0100λ≦G≦0.0695λ  (21)

FIG. 42A is a graph illustrating the range of Ψwhich satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.020λ≦H≦0.025%. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.42A as a polygon, it can be shown as in FIG. 42B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 42B.When expressing the range of the polygon shown in FIG. 42B in anapproximate equation, it can be expressed by Equations (22) to (24).Ψ≦14.0×G/λ+43.16 provided that 0.0100λ≦G≦0.0695λ  (22)Ψ≧−45.0×G/λ+43.45 provided that 0.0100λ≦G≦0.0600λ  (23)Ψ≧367.368×G/λ+18.608 provided that 0.0600λ≦G≦0.0695λ  (24)

FIG. 43A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.025λ≦H≦0.030%. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.43A as a polygon, it can be shown as in FIG. 43B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 43B.When expressing the range of the polygon shown in FIG. 43B in anapproximate equation, it can be expressed by Equations (25) to (27).Ψ≦12.0×G/λ+43.25 provided that 0.0100λ≦G≦0.0695λ  (25)Ψ≧−50.0×G/λ+43.32 provided that 0.0100λ≦G≦0.0500λ  (26)Ψ≦167.692×G/λ+32.435 provided that 0.0500λ≦G≦0.0695λ  (27)

FIG. 44A is a graph illustrating the range of Ψ which satisfies therequirement |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.030λ≦H≦0.035%. Here, the range sandwiched by thestraight line connecting the plots indicating the maximum value of Ψ andthe broken line connecting the plots indicating the minimum value of Ψis the range which satisfies the above-described condition.

In the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.44A as a polygon, it can be shown as in FIG. 44B, and it can be saidthat β satisfies the above-described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 44B.When expressing the range of the polygon shown in FIG. 44B in anapproximate equation, it can be expressed by Equations (28) to (30).Ψ≦12.0×G/λ+43.35 provided that 0.0100λ≦G≦0.0695λ  (28)Ψ≧−45.0×G/λ+42.80 provided that 0.0100λ≦G≦0.0500λ  (29)Ψ≧186.667×G/λ+31.217 provided that 0.0500λ≦G≦0.0695λ  (30)

Next, the change in the second-order temperature coefficient β when theangle θ is altered, that is, the relationship between θ and thesecond-order temperature coefficient β, is shown in FIG. 45. Here, theSAW device used in the simulation is a quartz crystal substrate with cutangles and SAW propagation direction of (0, θ, and Ψ) in Euler anglerepresentation, with a groove depth G of 0.04λ, and the electrode filmthickness H is 0.02λ. With regard to a value at which the absolute valueof the second-order temperature coefficient β is smallest in theabove-described angle range is appropriately selected, based on the θsetting angle. In addition, with regard to η, it is 0.6383, inaccordance with Equation (8).

Under these conditions, from FIG. 45, which shows the relationshipbetween θ and the second-order temperature coefficient β, it can be seenthat, provided that θ is within the range of 117° or more to 142° orless, the absolute values of the second-order temperature coefficient βare within a range of 0.01 (ppm/° C.²). Therefore, it can be said that,with the above-described setting values, by fixing θ in the range of117°≦θ≦142°, it is possible to configure a SAW resonator 10 which hasgood frequency-temperature characteristics.

Tables 17 to 19 are shown as simulation data proving the relationshipbetween θ and the second-order temperature coefficient β.

TABLE 17 H/λ G/λ θ β % % ° ppm/° C.² 0.01 4.00 117 −0.09 × 10⁻¹ 0.014.00 142   0.05 × 10⁻¹ 3.50 4.00 117 −0.09 × 10⁻¹ 3.50 4.00 142 −0.08 ×10⁻¹

Table 17 being a table illustrating the relationship between θ and thesecond-order temperature coefficient β when the electrode film thicknessH is changed, it shows the values of the second-order temperaturecoefficient β at the critical values (117° and 142°) of θ when theelectrode film thickness H is 0.01% λ, and when the electrode filmthickness H is 3.50% λ. The groove depths G in the simulation are all 4%λ. From Table 17, it can be seen that, in the range of 117°≦θ≦142°, eventhough the thickness of the electrode film thickness H is changed(0≅0.01% λ and 3.5% λstipulated as critical values of the electrode filmthickness), |β|≦0.01 (ppm/° C.²) is satisfied regardless of thethickness.

TABLE 18 H/λ G/λ θ β % % ° ppm/° C.² 2.00 1.00 117 −0.09 × 10⁻¹ 2.001.00 142 −0.08 × 10⁻¹ 2.00 6.95 117 −0.09 × 10⁻¹ 2.00 6.95 142 −0.09 ×10⁻¹

Table 18 being a table illustrating the relationship between θ and thesecond-order temperature coefficient β when the groove depth G ischanged, it shows the values of the second-order temperature coefficientβ at the critical values (117° and 142°) of θ when the groove depth G is1.00% λ and 6.95% λ. The electrode film thicknesses H in the simulationare all 2.00% λ. From Table 18, it can be seen that, in the range of117°≦θ≦142°, even though the groove depth G is changed (1.00% λ, and6.95% λstipulated as critical values of the groove depth G), |β|≦0.01(ppm/° C.²) is satisfied regardless of the depth.

TABLE 19 H/λ G/λ θ β % % η ° ppm/° C.² 2.00 4.00 0.62 117 −0.10 × 10⁻¹2.00 4.00 0.62 142 −0.03 × 10⁻¹ 2.00 4.00 0.76 117 −0.09 × 10⁻¹ 2.004.00 0.76 142 −0.09 × 10⁻¹

Table 19 being a table illustrating the relationship between θ and thesecond-order temperature coefficient β when the line occupation rate ηis changed, it shows the values of the second-order temperaturecoefficient β at the critical values (117° and 142°) of θ when the lineoccupation rate η is 0.62 and 0.76. The electrode film thicknesses H inthe simulation are all 2.00% λ, and the groove depths G are all 4.00% λ.From Table 19, it can be seen that, in the range of 117°≦θ≦142°, eventhough the line occupation rate η is changed (η=0.62 and 0.76 are theminimum value and maximum value of 1 when the groove depth is 4% λ inFIG. 31A, which shows the relationship between the line occupation rateη (η1) and groove depth G with the electrode film thickness H in therange of 0.020λ, to 0.025λ), |β|≦0.01 (ppm/° C.²) is satisfiedregardless of the value.

FIG. 46 is a graph illustrating the relationship between the angle φ andthe second-order temperature coefficient β in a case of using a quartzcrystal substrate 30 of (φ, 123°, and 43.77°) in Euler anglerepresentation, with a groove depth G of 0.04λ, an electrode filmthickness H of 0.02λ, and a line occupation rate η of 0.65.

From FIG. 46, it can be seen that when φ is −2° and +2°, thesecond-order temperature coefficient β is lower than −0.01 (ppm/° C.²)in each case, but provided that φ is in the range of −1.5° to +1.5°, theabsolute values of the second-order temperature coefficient β areconsistently a range of less than 0.01 (ppm/° C.²). Therefore, with theabove-described kinds of setting value, by fixing φ in the range of−1.5°≦φ+1.5°, or preferably −1°≦φ≦+1°, it is possible to configure a SAWresonator 10 which has good frequency-temperature characteristics.

In the above description, the range of optimum values of each of φ, θ,and Ψ is derived for the relationship with the groove depth G underspecific conditions. In contrast to this, an extremely preferablerelationship between θ and Ψ, where the frequency fluctuation amount at−40° C. to +85° C. is smallest, is shown in FIG. 47, and an approximateequation is obtained. According to FIG. 47, the angle Ψ changes alongwith an increase of the angle θ, increasing so as to describe a cubiccurve. In the example of FIG. 47, Ψ is 42.79° when θ=117°, and Ψ is49.57° when θ=142°. When illustrating these plots as an approximatecurve, they become the curve shown by the broken line in FIG. 47, andcan be expressed as an approximate equation by Equation (31).Ψ=1.19024×10⁻³×θ³−4.48775×10⁻¹×θ²+5.64362×10¹×θ−2.32327×10³±1.0  (31)

Because of this, it is possible to fix Ψ by θ being fixed, and it ispossible to make the range of Ψ 42.79°≦Ψ≦49.57° when the range of θ is117°≦θ≦142°. The groove depth G and electrode film thickness H in thesimulation are G=0.04λ, and H=0.02λ.

Due to the above-described kinds of reason, by configuring the SAWresonator 10 in accordance with the various conditions fixed in theembodiment, it is possible to obtain a SAW resonator which can realizegood frequency-temperature characteristics fulfilling the target values.

In addition, with the SAW resonator 10 according to the embodiment, theimprovement in the frequency-temperature characteristics is sought byhaving the film thickness H of the electrode film in the range of0<H≦0.035λ, as shown in Equation (7) and FIGS. 15A to 26F. This,differing from the improvement in the frequency-temperaturecharacteristics by making the film thickness H extremely large such asin the related art, realizes the improvement in thefrequency-temperature characteristics while still maintainingenvironmental resistance characteristics. The relationship between theelectrode film thickness (Al electrode film thickness) and frequencyfluctuation in a heat cycle test is shown in FIG. 54. The results of theheat cycle test shown in FIG. 54 are taken after a cycle in which theSAW resonator is exposed to an atmosphere of −55° C. for 30 minutes,then exposed for 30 minutes with the atmosphere temperature raised to+125° C., is repeated eight times. From FIG. 54, it can be seen that thefrequency fluctuation (F fluctuation) in the range of the electrode filmthickness H of the SAW resonator 10 according to the embodiment is ⅓ orless in comparison with that of a case where the electrode filmthickness H is 0.06λ, and where no inter-electrode finger groove isprovided. For every plot in FIG. 54, H+G=0.06λ.

In addition, on carrying out a high temperature storage test on a SAWresonator manufactured under the same conditions as those in FIG. 54,exposing it to an atmosphere of 125° C. for 1,000 hours, it is confirmedthat the frequency fluctuation amount of the SAW resonator according tothe embodiment before and after testing (the four conditions H=0.03λ andG=0.03λ, H=0.02λ and G=0.04λ, H=0.015λ and G=0.045λ, and H=0.01λ andG=0.05λ) is ⅓ or less in comparison with that of the SAW resonator(H=0.06× and G=0) in the related art.

In the SAW resonator 10 manufactured under the above-describedconditions, conditions where H+G=0.067λ (aluminum film thickness 2,000angstrom, groove depth 4,700 ANGSTROM), the IDT line occupation rateηi=0.6, the reflector line occupation rate ηr=0.8, the Euler angles are(0°, 123°, and 43.5°), the IDT pair number is 120, the intersectionwidth is 40λ (λ=10 μm), the number of reflectors (one side) is 72 (36pairs), and the electrode fingers have no angle of tilt (the electrodefinger array direction and SAW phase velocity direction correspond witheach other), frequency-temperature characteristics like those shown inFIG. 48 are exhibited.

In FIG. 48, the frequency-temperature characteristics are plotted forn=4 test specimens. According to FIG. 48, it can be seen that thefrequency fluctuation amount ΔF in the operating temperature range ofthe test specimens is suppressed to 20 ppm or less.

In the embodiment, a description has been made of the effect on thefrequency-temperature characteristics of the groove depth G, electrodefilm thickness H, and the like. However, the combined depth of thegroove depth G and electrode film thickness H (the level difference)also has an effect on static characteristics such as the equivalentcircuit constant and CI value, and on the Q value. For example, FIG. 49is a graph illustrating the relationship between the level differenceand CI value when changing the level difference from 0.062λ to 0.071λ.According to FIG. 49, it can be seen that the CI value converges whenthe level difference is 0.067λ, and does not improve (does not becomelower) even when increasing the level difference further.

The frequency, equivalent circuit constant, and static characteristicsof the SAW resonator 10 exhibiting frequency-temperature characteristicslike those shown in FIG. 48 are collected in FIG. 50. Here, F representsthe frequency, Q represents the Q value, γ represents the capacitanceratio, CI represents the CI (Crystal Impedance) value, and M representsthe figure of merit.

In addition, FIG. 52 shows a graph for comparing the relationshipbetween the level difference and Q value for the SAW resonator in therelated art and the SAW resonator 10 according to the embodiment. InFIG. 52, the graph shown with the thick line indicates thecharacteristics of the SAW resonator 10 according to the embodiment,grooves are provided between the electrode fingers, and the stop bandupper end mode resonance is used. The graph shown with the thin lineindicates the characteristics of the SAW resonator in the related art,the stop band upper end mode resonance is used with no groove beingprovided between the electrode fingers. As is clear from FIG. 52, whenproviding the grooves between the electrode fingers, and using the stopband upper end mode resonance, a higher Q value is obtained in the areain which the level difference (G+H) is 0.0407λ (4.07% λ) or more than ina case of using the stop band lower end mode resonance with no groovebeing provided between the electrode fingers.

The basic data of the SAW resonators according to the simulation is asfollows.

Basic data of SAW resonator 10 according to the embodiment

H: 0.02λ

G: changes

IDT line occupation rate ηi: 0.6

Reflector line occupation rate ηr: 0.8

Euler angles (0°, 123°, and 43.5°)

Pair number: 120

Intersection width: 40λ (λ=10 μm)

Reflector number (one side): 60

No electrode finger tilt angle

Basic data of SAW resonator in the related art

H: changes

G: zero

IDT line occupation rate ηi: 0.4

Reflector line occupation rate ηr: 0.3

Euler angles (0°, 123°, and 43.5°)

Pair number: 120

Intersection width: 40λ (λ=10 μm)

Reflector number (one side): 60

No electrode finger tilt angle

When referring to FIGS. 50 and 52 in order to compare thecharacteristics of the SAW resonators, it can be understood how muchhigher the Q of the SAW resonator 10 according to the embodiment is. Itcan be supposed that this high Q is due to an energy confinement effect,for the following reasons.

In order to efficiently confine the energy of a surface acoustic waveexcited in the stop band upper end mode, a stop band upper end frequencyft2 of the IDT 12 may be set between a stop band lower end frequency fr1of the reflectors 20 and a stop band upper end frequency fr2 of thereflectors 20, as in FIG. 53. That is, it may be set so as to satisfythe relationship offr1<ft2<fr2  (32)

With this, a reflection coefficient Γ of the reflectors 20 at the stopband upper end frequency ft2 of the IDT 12 increases, and the stop bandupper end mode SAW excited with the IDT 12 is reflected by thereflectors 20 to the IDT 12 side with a high reflection coefficient.Then, the stop band upper end mode SAW energy confinement becomesstronger, and it is possible to realize a low-loss resonator.

As opposed to this, in a case where the relationship between the stopband upper end frequency ft2 of the IDT 12 and the stop band lower endfrequency fr1 of the reflectors 20 and stop band upper end frequency fr2of the reflectors 20 is set to the condition of ft2<fr1, or thecondition of fr2<ft2, the reflection coefficient Γ of the reflectors 20at the stop band upper end frequency ft2 of the IDT 12 decreases, and itbecomes difficult to realize a strong energy confinement condition.

Here, in order to realize the condition of Equation (32), it isnecessary to make a frequency shift of the stop band of the reflectors20 to an area higher than the stop band of the IDT 12. Specifically,this can be realized by making the array cycle of the conductor strips22 of the reflectors 20 shorter than that of the array cycle of theelectrode fingers 18 of the IDT 12. In addition, as in other methods, itcan be realized by making the thickness of the electrode film formed asthe conductor strips 22 of the reflectors 20 less than the thickness ofthe electrode film formed as the electrode fingers 18 of the IDT 12, orby making the depth of the inter-conductor strip grooves of thereflectors 20 less than the depth of the inter-electrode finger groovesof the IDT 12. In addition, a plurality of these methods may be employedin combination.

According to FIG. 50, it can be said that, apart from the high Q, it ispossible to obtain a high figure of merit M. In addition, FIG. 51 is agraph illustrating the relationship between an impedance Z and thefrequency for the SAW resonator from which FIG. 50 is obtained. FromFIG. 51, it can be seen that no unnecessary spurious response exists inthe vicinity of the resonance point.

As described above, as the SAW resonator according to the embodiment ofthe invention has inflection points within the operating temperaturerange (temperature range to be used: −40° C. to +85° C.), as shown inFIG. 48, it is possible to realize frequency-temperature characteristicswhich describe a cubic curve, or an approximate cubic curve, with anextremely small frequency fluctuation amount of approximately 20 ppm orless.

FIG. 56A is a graph illustrating the frequency-temperaturecharacteristics of the SAW resonator disclosed in JP-A-2006-203408.Although the frequency-temperature characteristics describe a cubiccurve, as an inflection point exists in an area beyond the operatingtemperature range (temperature range to be used: −40° C. to +85° C.), ascan be seen, it is essentially a quadratic curve which has an upwardlyconvex peak, as shown in FIG. 56B. For this reason, the frequencyfluctuation amount has an extremely high value of 100 (ppm).

As opposed to this, the SAW resonator according to the embodiment of theinvention, with the frequency fluctuation amount describing a cubiccurve, or an approximate cubic curve, within the operating temperaturerange, realizes a dramatic reduction of the frequency fluctuationamount. Changes in the frequency fluctuation amount within the operatingrange for a SAW resonator whose IDT and reflectors are covered with aprotective film are shown in FIGS. 57 and 58.

The example shown in FIG. 57 is a diagram illustrating the frequencyfluctuation amount within the operating temperature range when theelectrodes are coated with alumina as a protective film. According toFIG. 57, it can be seen that it is possible to keep the frequencyfluctuation amount within the operating temperature range at 10 (ppm) orless.

Basic data of SAW resonator according to example shown in FIG. 57

H: (material: aluminum): 2,000 (ANGSTROM)

G: 4700 (ANGSTROM)(H+G=0.067λ)IDT line occupation rate ηi: 0.6Reflector line occupation rate ηr: 0.8In-plane rotation ST cut substrate with Euler angles (0°, 123°, and43.5°)Pair number: 120Intersection width: 40λ (λ=10 μm)Reflector number (one side): 36No electrode finger tilt angleProtective film (alumina) thickness 400 (ANGSTROM)Second-order temperature coefficient β=+0.0007 (ppm/° C.²)

The example shown in FIG. 58 is a diagram illustrating the frequencyfluctuation amount within the operating temperature range when theelectrodes are coated with SiO₂ as a protective film. According to FIG.58, it can be seen that it is possible to keep the frequency fluctuationamount within the operating temperature range at 20 (ppm) or less.

Basic data of SAW resonator according to example shown in FIG. 58

H: (material: aluminum): 2,000 (ANGSTROM)

G: 4,700 (ANGSTROM)(H+G=0.067λ)IDT line occupation rate ηi: 0.6Reflector line occupation rate ηr: 0.8In-plane rotation ST cut substrate with Euler angles (0°, 123°, and43.5°)Pair number: 120Intersection width: 40λ (λ=10 μm)Reflector number (one side): 36No electrode finger tilt angleProtective film (SiO₂) thickness 400 (ANGSTROM)Second-order temperature coefficient β=+0.0039 (ppm/° C.²)

The inventor has described that the absolute value of the second-ordertemperature coefficient β can be made 0.01 ppm/° C.² or less byadjusting the line occupation rate η for the above-described designed Gand H. On the other hand, the inventor has found that, in a case wherethe line occupation rate η fluctuates when excitation is performed inthe stop band upper end mode, the frequency-temperature characteristicsof the surface acoustic wave fluctuate. When a plurality of surfaceacoustic wave resonators are manufactured, there are cases in which itis difficult to match all the surface acoustic wave resonators with adesired design line occupation rate and thus surface acoustic waveresonators with a line occupation rate different from the design lineoccupation rate may be manufactured. At this time, fluctuation amountsof the line occupation rates with the design line occupation rate arenot constant, and variations in the line occupation rates occur.Therefore, when a plurality of surface acoustic wave resonators aremanufactured, variations occur in the frequency-temperaturecharacteristics of the surface acoustic wave. The inventor has foundthat this causes variations in frequency deviations in an operatingtemperature range of the surface acoustic wave element, which is a causeof reducing a yield of surface acoustic wave resonators.

In addition, as described above, the inventor has found that a value ofthe line occupation rate η when a fluctuation amount of the first-ordertemperature coefficient is the minimum in a case where variations occurin the line occupation rate η and a value of the line occupation rate ηwhere the second-order temperature coefficient β is the minimum(−0.01≦β≦0.01) do not correspond with each other. Therefore, in thesecond embodiment, a description will be made of a SAW device which cansuppress a value of the second-order temperature coefficient β to theabove-described range, and can minimize a fluctuation amount of thefirst-order temperature coefficient which dominantly functions in thefrequency-temperature characteristics in the operating temperaturerange.

FIGS. 59A to 59D are schematic diagrams of a SAW resonator according tothe second embodiment. In FIGS. 59A to 59D, FIG. 59A is a plan view ofthe SAW resonator according to the second embodiment, FIG. 59B is apartial enlarged sectional view, FIG. 59C is an enlarged view fordescribing details of FIG. 59B, and FIG. 59D is a diagram which, relatedto the partial enlarged view of FIG. 59C, is for describing an IDTelectrode finger effective line occupation rate ηeff identificationmethod in a case where the cross-sectional shape is not rectangular buttrapezoidal, which is a conceivable sectional shape when the SAWresonator according to the embodiment of the invention is manufacturedusing a photolithography technique and an etching technique.

A basic configuration of the SAW resonator according to the secondembodiment is similar to that of the surface acoustic wave resonator 10according to the first embodiment. That is to say, the SAW resonator 110includes a quartz crystal substrate 30, an IDT 112 formed on the quartzcrystal substrate, and reflectors 120 formed at both sides of the IDT112. The IDT 112 extends in a direction perpendicular to the directionwhere surface acoustic wave (wavelength λ) propagates, and is formed bypectinate electrodes 114 a and 114 b which intersect each other. Thepectinate electrode 114 a is constituted by a plurality of electrodefingers 118 a which are arranged in the propagation direction of thesurface acoustic wave at the same interval (λ) and a bus bar 116 a whichconnects a plurality of electrode fingers 118 a in parallel to eachother. In a similar way, the pectinate electrode 114 b is constituted bya plurality of electrode fingers 118 b which are arranged in thepropagation direction of the surface acoustic wave at the same interval(λ) and a bus bar 116 b which connects a plurality of electrode fingers118 b in parallel to each other.

Therefore, the electrode fingers 118 a and the electrode fingers 118 bare alternately disposed in the propagation direction of the surfaceacoustic wave at the same interval (λ/2). The reflectors 120 haveconductor strips 122 which are disposed in the propagation direction ofthe surface acoustic wave at the same interval (λ/2). In addition, theSAW resonators 110 excite the surface acoustic wave in the stop bandupper end mode.

In the first embodiment, the widths of the convex portions 34 formed bythe inter-electrode finger grooves 32 correspond with those of theelectrode fingers 18 a and 18 b in the propagation direction of thesurface acoustic wave. However, the second embodiment is different fromthe first embodiment in that the widths of the electrode fingers 118 aand 118 b in the propagation direction of the surface acoustic wave onthe convex portions 134 formed by the inter-electrode finger grooves 132are smaller than the widths of the convex portions in the propagationdirection of the surface acoustic wave. In addition, both ends of theelectrode fingers 118 a and 118 b in the propagation direction of thesurface acoustic wave are disposed at inner sides than both ends of theconvex portions 134 in the propagation direction of the surface acousticwave in a plan view. Therefore, both the ends of the convex portions 134in the propagation direction of the surface acoustic wave are notcovered with the electrode fingers 118 a and 118 b and thus are exposed.

In addition, the same configuration is formed in the reflectors 120which are disposed at both sides of the IDT 112.

In FIG. 59C, when the width of the convex portion 134 in the propagationdirection of the surface acoustic wave is Lg, and the width of each ofthe electrode fingers 118 a and 118 b in the propagation direction ofthe surface acoustic wave is Le, it givesL _(g) >L _(e)  (33).

With this, since an amount (solid angle) where lines of electric forceemitted from the electrode fingers 118 a and 118 b are absorbed by theconvex portions 134 increases, an excitation efficiency of the surfaceacoustic wave increases, thereby reducing loss in the SAW resonator 110in comparison with the first embodiment.

However, in the SAW resonator 10 according to the first embodiment, thesurface acoustic wave is reflected at the parts where step differencesof both ends of the convex portion 34 (refer to FIGS. 1A to 1D) with thewidth L in the thickness (G) direction rise. In addition, the lineoccupation rate η is adjusted by adjusting the width L, and thus it ispossible to minimize the second-order temperature coefficient β byadjusting a reflection position of the surface acoustic wave. Further, afluctuation amount of the first-order temperature coefficient depends onthe line occupation rate η of the convex portions 34, and thus it ispossible to minimize the fluctuation amount of the first-ordertemperature coefficient by adjusting a value thereof.

However, since the widths of the convex portions 134 corresponding tothe line occupation rate η are also L, in the configuration according tothe first embodiment, it is difficult to simultaneously specify a lineoccupation rate η which minimizes the second-order temperaturecoefficient β and a line occupation rate η which minimizes a fluctuationamount of the first-order temperature coefficient.

On the other hand, in the SAW resonator 110 according to the secondembodiment, the surface acoustic wave is reflected at the parts wherestep differences of both ends of the convex portion 134 with the widthLg in the thickness (G) direction rise and at the parts where stepdifferences of both ends of each of the electrode fingers 118 a and 118b with the width Le in the thickness (H) direction rise. Therefore, inthe second embodiment, it is supposed that the electrode fingers 118 aand 118 b haveL _(eff)=(L _(g) +L _(e))/2  (34)as an effective width and reflect the surface acoustic wave at the bothend positions.

Therefore, if a line occupation rate of the convex portions 134 is ηg(=Lg/P), and a line occupation rate of the electrode fingers 118 a and118 b is ηe (=Le/P), the effective line occupation rate ηeff at thistime becomesη_(eff)=(η_(g)+η_(e))/2  (35).

At this time,η_(g)>η_(e)  (36).Therefore, the second-order temperature coefficient can be adjusted byadjusting ηeff. On the other hand, a fluctuation amount of thefirst-order temperature coefficient can be adjusted using ηg. Thus, thesecond-order temperature coefficient can be adjusted by adjusting ηeff,and a fluctuation amount of the first-order temperature coefficient canbe adjusted by adjusting ηg.

Further, in a case where, as in FIG. 59D, both side surfaces ofelectrode fingers 119 and convex portions 135 in the width direction aretilted, and the cross-sectional shapes thereof are trapezoidal, it isassumed that the lower end width of the electrode finger 119 in thethickness direction is Leb, and the upper end width thereof is Let. Inaddition, it is assumed that the lower end width of the convex portion135 in the thickness direction is Lgb, and the upper end width thereofis Lgt. At this time,

$\begin{matrix}\left\{ {\begin{matrix}{L_{eb} > L_{et}} \\{L_{gb} > L_{gt}} \\{L_{gt} > L_{eb}}\end{matrix}.} \right. & (37)\end{matrix}$

Here, Le and Lg are defined as

$\begin{matrix}\left\{ {\begin{matrix}{L_{e} = \frac{L_{et} + L_{eb}}{2}} \\{L_{g} = \frac{L_{gt} + L_{gb}}{2}}\end{matrix}.} \right. & (38)\end{matrix}$

The inventor investigated a frequency deviation, a CI value, and afluctuation amount of the first-order temperature coefficient in a case(type 1) where the width Le of the electrode finger corresponds with thewidth Lg of the convex portion as in the first embodiment and in a case(type 2) of Lg>Le as in the second embodiment.

In the SAW device used in the investigation for both of the types 1 and2, the Euler angles were (φ=0°, θ=123°, and) Ψ=44°), G=0.046λ, H=0.021λ,the IDT electrode pair number was 210, and the number of the reflectorsdisposed at both ends of the IDT electrodes was 94 at one sides.

First, as an example 1, in the type 1, the line occupation rate ηe ofthe electrode fingers was 0.64, and the line occupation rate ηg of theconvex portions 34 (refer to FIGS. 1A to 1D) was 0.64 such that the lineoccupation rate η was 0.64. In the type 2, the ηe was 0.57 and ηg was0.71 such that the effective line occupation rate ηeff was 0.64.

Next, as an example 2, in the type 1, the line occupation rate ηe of theelectrode fingers 118 and 118 b was 0.66, and the line occupation rateηg of the convex portions 134 was 0.66 such that the line occupationrate η was 0.66. In the type 2, the ηe was 0.59 and ηg was 0.73 suchthat the effective line occupation rate ηeff was 0.66. That is to say,in both the examples, ηeff in the type 2 maintains the value of η in thetype 1, and is adjusted such that ηg>ηe. In addition, with regard to aCI value, a plurality of samples (1784) were prepared for each type, andan average value thereof was calculated.

FIG. 60 shows frequency-temperature characteristics of the type 1 andthe type 2 in the example 1, and FIG. 61 shows frequency-temperaturecharacteristics of the type 1 and the type 2 in the example 2. As shownin FIG. 60, in the example 1, both of the type 1 and the type 2 describea tertiary functional curve having about 25° C. as a temperature(reference temperature) of the inflection point. In addition, as shownin FIG. 61, in the example 2, both of the type 1 and the type 2 describea tertiary functional curve having about 40° C. as a temperature(reference temperature) of the inflection point. Therefore, it can beseen that both the type 1 and the type 2 have the very small value ofthe second-order temperature coefficient β in the examples 1 and 2.

As shown in FIG. 60, in the example 1, both the type 1 and the type 2have the frequency deviation of 12 ppm at −40° C. to +85° C. and thesame curve shape. In addition, as shown in FIG. 61, in the example 2,the type 1 has the frequency deviation of 18 ppm, and the type 2 has thefrequency deviation of 16 ppm. Therefore, it can be seen that the type 2maintains the frequency-temperature characteristics of the type 1 aslong as ηeff of the type 2 maintains η (ηg) of the type 1 even if thetype 1 is transformed into the type 2.

Further, in the example 1, the CI value of the type 1 was 23.8Ω, butthat of the type 2 was improved to 20.1Ω. In addition, in the example 2,the CI value of the type 1 was 22.4Ω, but that of the type 2 wasimproved to 19.2Ω, and thus the type 2 realizes a low-loss SAW resonatorin the examples 1 and 2. It is thought that this is because the solidangle at which the convex portions anticipate the lines of electricforce generated from the electrode fingers is greater in the type 2 thanin the type 1, and thus the excitation efficiency increases, asdescribed above.

Next, changes in the fluctuation amounts of the first-order temperaturecoefficient of the type 1 and the type 2 were investigated. FIG. 62shows a change of the fluctuation amount of the first-order temperaturecoefficient when the line occupation rate η of the type 1 changes by0.01, and FIG. 63 shows a change of the fluctuation amount of thefirst-order temperature coefficient when the effective line occupationrate ηeff of the type 2 changes by 0.01. In the SAW device used in theinvestigation for both of the type 1 and the type 2, the Euler angleswere (φ=0°, θ=123°, and Ψ=44°), G=0.045λ, and H=0.02λ.

As shown in FIG. 62, in the type 1, the fluctuation amount of thefirst-order temperature coefficient shows monotone decreasing if η (orηg) increases from 0.60, it has the minimal value at 0.70, and thenshows monotone increasing at 0.70 or more.

On the other hand, as shown in FIG. 63, in the type 2, the fluctuationamount of the first-order temperature coefficient shows monotonedecreasing if ηeff increases from 0.53, it has the minimal value at0.63, and then shows monotone increasing at 0.63 or more.

As described above, the type 1 and the type 2 in the example 1 have afavorable frequency deviation of 12 ppm as shown in FIG. 60. However, inthe type 1, the fluctuation amount of the first-order temperaturecoefficient is 0.6 ppm/° C. when η (or ηg) is 0.64 as shown in FIG. 62.Therefore, due to the variation in the line occupation rate η (ηg),variations in the first-order temperature coefficient (that is,variations in the frequency-temperature characteristics) is remarkable.On the other hand, it can be seen that, in the type 2, if an approximatecurve is generated from the plot in FIG. 63 and ηeff=0.64 is extractedfrom the approximate curve, the fluctuation amount of the first-ordertemperature coefficient is about 0.2. Further, from FIG. 63, it can beseen that the following relationship can suppress the fluctuation amountof the first-order temperature coefficient to 0.6 ppm/° C. or less.0.58<η_(eff)<0.73  (39)

In addition, a plurality of samples (1784) for the type 1 and the type 2in the example 1 were prepared, and variations in the first-ordertemperature coefficient and variations in the resonance frequency at 25°C. were investigated. If the magnitude of the variation in thefirst-order temperature coefficient of the type 1 is 1, the magnitude ofthe variation in the first-order temperature coefficient of the type 2is improved up to 0.2. As described above, in the type 1, the lineoccupation rate ηg of the convex portions 34 (refer to FIGS. 1A to 1D)is 0.64, where the fluctuation amount of the first-order temperaturecoefficient is 0.6 ppm/° C. from FIG. 62. On the other hand, in the type2, it is confirmed that if the line occupation rate ηg of the convexportions 134 is 0.71, an approximate curve is generated from the plot inFIG. 62, and ηg=0.71 is extracted from the approximate curve, thefluctuation amount of the first-order temperature coefficient is about0.1 ppm/° C. In addition, it is considered that this improvement can beseen since the variation in the first-order temperature coefficient isproportional to the fluctuation amount of the first-order temperaturecoefficient. Further, in relation to the variation in the resonancefrequency at 25° C. as well, if the magnitude thereof is 1 in the type1, the magnitude thereof is improved up to 0.5 in the type 2.

In addition, from the above description, the type 2 maintains thefrequency-temperature characteristics of the type 1 which has η equal toa value of ηeff. Therefore, it may be considered that η is replaced withηeff in all the drawings where η is expressed by the above-describedtransverse axis. Therefore, ηeff is designed so as to be included in therange of Equation (39) and the range which is surrounded by the lineconnected so as to make a round in an alphabetical order, that is, therange satisfying the above Equations (3), (5) and (6) in FIG. 13B, tothereby suppress the fluctuation amount in the first-order temperaturecoefficient, thereby suppressing the variations in the frequencydeviation in the operating temperature range of the SAW resonator 110,and suppressing an absolute value of the second-order temperaturecoefficient β to 0.01 (ppm/° C.²) or less.

Further, ηeff is set so as to be included in the range of Equation (39),and a range included in the range surrounded by the line connecting therespective points shown with the plane coordinates (G/λ, ηeff) in thefigures in an alphabetical order so as to make a round, in FIG. 27A(0<H≦0.005λ), FIG. 28A (0.005λ<H≦0.010λ), FIG. 29A (0.010λ<H≦0.015λ),FIG. 30A (0.015λ<H≦0.020λ), FIG. 31A (0.020λ<H≦0.025λ), FIG. 32A(0.025λ<H≦0.030λ), and FIG. 33A (0.030λ<H≦0.035λ). This can suppress thevariations in the first-order temperature coefficient, therebysuppressing the variations in the frequency-temperature characteristics,that is, the variations in the frequency deviation in the operatingrange of the SAW resonator 110, and suppressing the absolute value ofthe second-order temperature coefficient β to 0.01 (ppm/° C.²) or lessso as to correspond to the film thickness H of the electrode finger.

In the embodiments, the IDT 12 forming the SAW resonator 10 and the IDT112 forming the SAW resonator 110 are shown such that all the electrodefingers alternately intersect each other. However, the SAW resonator 10and the SAW resonator 110 according to the embodiments of the inventioncan achieve considerable effects only with the quartz crystal substrate.For this reason, even in a case where the electrode fingers 18 in theIDT 12 and the electrode fingers 118 a and 118 b in the IDT 112 arethinned out, it is possible to achieve the same effect.

In addition, in the first embodiment, the grooves 32 may be partiallyprovided between the electrode fingers 18 and between the conductorstrips 22 of the reflectors 20. In particular, as the central portion ofthe IDT 12, which has a high oscillatory displacement, has a dominanteffect on the frequency-temperature characteristics, a structure may beadopted where the grooves 32 are provided only in that portion. In thiskind of structure as well, it is possible to achieve a SAW resonator 10with good frequency-temperature characteristics.

In addition, in the above-described embodiment, it is noted that Al oran Al-based alloy is used as the electrode film. However, the electrodefilm may be configured using another metal material, provided that it isa metal which can achieve the same effect as the embodiment. Further, aprotective film such as SiO₂ or alumina may be provided on the electrodefilm.

In addition, although the above-described embodiment is a one-terminalpair SAW resonator in which only one IDT is provided, the invention isalso applicable to a two-terminal pair SAW resonator in which aplurality of IDTs are provided, and is also applicable to alongitudinally coupled or transversally coupled double mode SAW filteror multiple mode SAW filter.

Next, a description will be made, referring to FIGS. 55A and 55B, of theSAW oscillator according to the embodiment of the invention. The SAWoscillator according to the embodiment of the invention, as shown inFIGS. 55A and 55B, is configured of the SAW resonator 10 (the SAWresonator 110), an IC (integrated circuit) 50, which applies voltage tothe IDT 12 of the SAW resonator 10 (the SAW resonator 110) so as tocontrol driving thereof, and a package which houses them. In FIGS. 55Aand 55B, FIG. 55A is a plan view with a lid removed, and FIG. 55B is across-sectional view taken along the line A-A in FIG. 55A.

In the SAW oscillator 100 according to the embodiment, the SAW resonator10 and IC 50 are housed in the same package 56, and electrode patterns54 a to 54 g formed on a bottom plate 56 a of the package 56, and thepectinate electrodes 14 a and 14 b of the SAW resonator 10 and pads 52 ato 52 f of the IC 50, are connected by metal wires 60. Then, a cavity ofthe package 56 housing the SAW resonator 10 (the SAW resonator 110) andIC 50 is hermetically sealed with a lid 58. By adopting thisconfiguration, it is possible to electrically connect the IDT 12 (referto FIGS. 1A to 1D), the IC 50, and an externally mounted electrode (notshown) formed on the bottom surface of the package 56.

Therefore, in response to a demand for an expansion of operatingtemperature range and higher accuracy of internally mounted electronicdevices, with the effect of internal heat generation increasing alongwith the miniaturization of blade servers and other packages, inaddition to a higher reference clock frequency due to the speeding-up ofinformation communication in recent years, and furthermore, in responseto a market which needs long-term, stable operating in environments fromlow temperature to high temperature, such as wireless base stationsinstalled outdoors, the SAW oscillator according to the embodiments ofthe invention is preferred, as it has extremely goodfrequency-temperature characteristics of a frequency fluctuation amountof approximately 20 (ppm) or less in its operating temperature range(temperature range to be used: −40° C. to +85° C.)

Furthermore, as the SAW resonator according to the invention, or SAWoscillator including the SAW resonator, realizes a significantimprovement in frequency-temperature characteristics, it contributeslargely to realizing a product with, as well as extremely goodfrequency-temperature characteristics, excellent jitter characteristicsand phase noise characteristics, for example, a mobile telephone, a harddisk, a personal computer, a tuner receiving a BS and CS broadcast, aninstrument processing a high frequency signal transmitted through acoaxial cable or an optical signal transmitted through an optical cable,or an electronic instrument such as a server network instrument orwireless communication instrument which needs a high frequency, highaccuracy clock (low jitter, low phase noise) in a wide temperaturerange, and it goes without saying that it contributes greatly to furthersystem reliability and quality improvement.

What is claimed is:
 1. A surface acoustic wave resonator comprising: aquartz crystal substrate with Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°,and 42.79°≦|Ψ|≦49.57°); an IDT that is provided on the quartz crystalsubstrate, that includes a plurality of electrode fingers, and thatexcites a stop band upper end mode surface acoustic wave; andinter-electrode finger grooves that are provided in the quartz crystalsubstrate between the electrode fingers in a plan view, wherein when awavelength of the surface acoustic wave is λ, a first film thickness ofthe electrode finger is H, and a first depth of the inter-electrodefinger groove is G, and when a line occupation rate of convex portionsof the quartz crystal substrate disposed between the inter-electrodefinger grooves is ηg, and a line occupation rate of the electrodefingers disposed on the convex portions is ηe, the followingrelationships are satisfied:0.0407λ≦G+H; andηg>ηe.
 2. The surface acoustic wave resonator according to claim 1,further comprises: reflectors that sandwich the IDT in a propagationdirection of the surface acoustic wave and that have a plurality ofconductor strips.
 3. The surface acoustic wave resonator according toclaim 2, wherein when a stop band upper end mode frequency of the IDT isft2, a stop band lower end mode frequency of the reflectors is fr1, anda stop band upper end mode frequency of the reflectors is fr2, thefollowing relationship is satisfied:fr1<ft2<fr2.
 4. The surface acoustic wave resonator according to claim2, further comprises: inter-conductor strip grooves that are provided inthe quartz crystal substrate between the plurality of conductor stripsin the plan view.
 5. The surface acoustic wave resonator according toclaim 3, further comprises: inter-conductor strip grooves that areprovided in the quartz crystal substrate between the plurality ofconductor strips in the plan view.
 6. The surface acoustic waveresonator according to claim 5, wherein a second depth of theinter-conductor strip groove is smaller than the first depth of theinter-electrode finger groove.
 7. The surface acoustic wave resonatoraccording to claim 5, wherein a second film thickness of the conductorstrip is thinner than the first film thickness of the electrode finger.8. The surface acoustic wave resonator according to claim 1, wherein theplurality of electrode fingers are made of aluminum or a metal alloywith aluminum, aluminum is a main element of the metal alloy.
 9. Thesurface acoustic wave resonator according to claim 2, wherein theplurality of electrode fingers and the conductor strips are made ofaluminum or a metal alloy with aluminum, aluminum is a main element ofthe metal alloy.
 10. The surface acoustic wave resonator according toclaim 3, wherein the plurality of electrode fingers and the conductorstrips are made of aluminum or a metal alloy with aluminum, aluminum isa main element of the metal alloy.
 11. The surface acoustic waveresonator according to claim 5, wherein the plurality of electrodefingers and the conductor strips are made of aluminum or a metal alloywith aluminum, aluminum is a main element of the metal alloy.
 12. Asurface acoustic wave oscillator comprising the surface acoustic waveresonator according to claim 1 and a circuit that drives the IDT.
 13. Asurface acoustic wave oscillator comprising the surface acoustic waveresonator according to claim 2 and a circuit that drives the IDT.
 14. Asurface acoustic wave oscillator comprising the surface acoustic waveresonator according to claim 3 and a circuit that drives the IDT.
 15. Asurface acoustic wave oscillator comprising the surface acoustic waveresonator according to claim 5 and a circuit that drives the IDT.
 16. Anelectronic apparatus comprising the surface acoustic wave resonatoraccording to claim
 1. 17. An electronic apparatus comprising the surfaceacoustic wave resonator according to claim
 2. 18. An electronicapparatus comprising the surface acoustic wave resonator according toclaim
 3. 19. An electronic apparatus comprising the surface acousticwave resonator according to claim 5.